{"id":8302,"date":"2026-03-19T16:45:37","date_gmt":"2026-03-19T16:45:37","guid":{"rendered":"https:\/\/mrenglishkj.com\/?p=8302"},"modified":"2026-03-26T02:59:23","modified_gmt":"2026-03-26T02:59:23","slug":"sat-math-module-2nd-how-to-get-1500-hack-free-test-2024","status":"publish","type":"post","link":"https:\/\/us.mrenglishkj.com\/sat\/sat-math-module-2nd-how-to-get-1500-hack-free-test-2024\/","title":{"rendered":"SAT Math Module 2nd (How to Get 1500+ Hack, Free Test 2024"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">The SAT 2024 Examination Test (Math Module 2nd with Step-By-Step Solutions, Tips and Desmos Tricks<\/h2>\n\n\n\n<p>How was your Module 1st? How much score have you made? Please tell us in the comment. The SAT math seems tough without Desmos Calculator but we have solution for this. This test is a practice test of 2024 SAT Math Module Second. Here, you would see questions that were possible to be on 2024 examination. The best parts are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>solutions of all questions,<\/li>\n\n\n\n<li>step-by-step explanations,<\/li>\n\n\n\n<li>how to verify the correct answer,<\/li>\n\n\n\n<li>description of correct and incorrect options,<\/li>\n\n\n\n<li>tips and tricks,<\/li>\n\n\n\n<li>and Desmos Calculator Hacks.<\/li>\n<\/ul>\n\n\n\n<p>Like the other exams, it has the same format and all the necessary features for you to become a SAT master in math. You just take the Module 2nd exam to practice your skills. The best part is that you practice within the time limit, and there are explanations of answers, tips and tricks to get a perfect score on the SAT. You will find Math easy after this.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">ABOUT THE SAT MODULES<\/h3>\n\n\n\n<p>The SAT is divided into four modules. There are two categories with each split into two modules. The first category is &#8220;Reading and Writing&#8221; with two modules. The second category is &#8220;Math&#8221; with two modules. The one, you will do below is SAT Math 2024 Practice Test Module 2nd.<\/p>\n\n\n\n<p>The first module has questions ranging from easy to difficult, but the second module only contains medium to difficult questions, no easy. If you want to take some other SATs, visit the links below.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-english\/module-1st\/\" target=\"_blank\" rel=\"noopener\" title=\"\">1st Module of SAT Reading And Writing Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-english\/module-2nd\/\" target=\"_blank\" rel=\"noopener\" title=\"\">2nd Module of SAT Reading And Writing Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-math\/1st-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">1st Module of SAT Math Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-math\/2nd-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">2nd Module of SAT Math Practice Tests<\/a><\/li>\n<\/ul>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">THE SAT MATH MODULE 2ND<\/h3>\n\n\n\n<p>The second module of Math in SAT contains four segments: &#8220;Algebra,&#8217; &#8216;Advanced Math,&#8217; &#8216;Problem-Solving and Data Analysis,&#8217; and &#8216;Geometry and Trigonometry.&#8221; The questions in Module 2nd are from medium to difficult. In a real SAT exam, you must answer 22 questions within 35 minutes. We have provided you with the same in this Practice Test.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Instructions for the SAT Real-Time Exam: Tips Before Taking Tests<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Previous-and-Next:<\/strong> Like in real SAT exam, you can move freely from one question to another, same things you can do here. You select one option and move forward but you realized something, so you came back and change your option. You can do that here and in the real SAT exam too.<\/li>\n\n\n\n<li><strong>Timer: <\/strong>On the top of the slide, you will see the timer, it starts from 0 and for Module 1st of Math you will get <strong><em>35 minutes to finish 22 questions<\/em><\/strong>. Always try to finish the test before 35 minutes.<\/li>\n\n\n\n<li><strong>Image:<\/strong> You can click on a graph, table, or other image to expand it and view it in full screen.<\/li>\n\n\n\n<li><strong>Mobile:<\/strong> You cannot take the real exam on mobile, but our practice exam you can take on mobile phone.<\/li>\n\n\n\n<li><strong>Calculator<\/strong>: Below the Test, you will see a Desmos calculator and graph for Math. The same, Desmos, will be used in real exams, so learn &#8220;How to use Desmos Calculator.&#8221;<\/li>\n\n\n\n<li><strong>Answer All<\/strong>: Even if you do not know the correct answer of a question, still guess it because there is no Negative marking.<\/li>\n\n\n\n<li><strong>Last Questions<\/strong>: The harder the question, the more marks it will fetch for you. So most likely, you will find later question difficult and more time-consuming, so utilize your time accordingly.<\/li>\n\n\n\n<li><strong>Tips:<\/strong> This article will help you learn more about the SAT Exams. <a href=\"https:\/\/us.mrenglishkj.com\/sat\/everything-about-the-sat\/\" target=\"_blank\" rel=\"noopener\" title=\"SAT: EVERYTHING ABOUT THE SAT\">SAT: EVERYTHING ABOUT THE SAT<\/a><\/li>\n<\/ol>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n        <script>\n          window.KQ_FRONT = window.KQ_FRONT || {};\n          window.KQ_FRONT.quiz_id = 4;\n          window.KQ_FRONT.rest = \"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/kq\/v1\/\";\n        <\/script>\n        <div id=\"kapil-quiz-4\"\n             class=\"kapil-quiz-container\"\n             data-kq-app\n             data-quiz-id=\"4\">\n            <div class=\"kq-loading\">Loading quiz...<\/div>\n        <\/div>\n        \n    <div id=\"kq-auth-modal\" class=\"kq-auth-modal\" style=\"display:none;\">\n      <div class=\"kq-auth-modal-inner\">\n        <button id=\"kq-auth-close\" class=\"kq-auth-close\" aria-label=\"Close\">\u2716<\/button>\n\n        <!-- TAB NAV -->\n        <div class=\"kq-auth-tabs\" role=\"tablist\">\n          <button class=\"kq-tab active\" data-tab=\"register\" type=\"button\" role=\"tab\" aria-selected=\"true\">Register<\/button>\n          <button class=\"kq-tab\" data-tab=\"login\" type=\"button\" role=\"tab\" aria-selected=\"false\">Login<\/button>\n          <button class=\"kq-tab\" data-tab=\"forgot\" type=\"button\" role=\"tab\" aria-selected=\"false\">Forgot<\/button>\n        <\/div>\n\n        <!-- PANELS -->\n        <div class=\"kq-auth-panel-wrap\">\n\n          <!-- REGISTER -->\n          <div class=\"kq-auth-panel\" data-panel=\"register\" style=\"display:block\">\n            <div class=\"kq-auth-card\">\n              <h3>Register<\/h3>\n              <div class=\"kq-field\">\n                <input id=\"kq-signup-username\" placeholder=\"Username\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-signup-email\" placeholder=\"Email\" type=\"email\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-signup-password\" placeholder=\"Password\" type=\"password\" \/>\n                <button class=\"kq-toggle-pass\" type=\"button\" aria-label=\"Toggle password\">\ud83d\udc41<\/button>\n              <\/div>\n              <button id=\"kq-signup-btn\" class=\"button kq-btn-small\">Register<\/button>\n              <small style=\"display:block;margin-top:8px;\">Already registered? Use Login tab.<\/small>\n            <\/div>\n          <\/div>\n\n          <!-- LOGIN -->\n          <div class=\"kq-auth-panel\" data-panel=\"login\" style=\"display:none\">\n            <div class=\"kq-auth-card\">\n              <h3>Login<\/h3>\n              <div class=\"kq-field\">\n                <input id=\"kq-login-identity\" placeholder=\"Username or Email\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-login-password\" placeholder=\"Password\" type=\"password\" \/>\n                <button class=\"kq-toggle-pass\" type=\"button\" aria-label=\"Toggle password\">\ud83d\udc41<\/button>\n              <\/div>\n              <button id=\"kq-login-btn\" class=\"button kq-btn-small\">Login<\/button>\n            <\/div>\n          <\/div>\n\n          <!-- FORGOT -->\n          <div class=\"kq-auth-panel\" data-panel=\"forgot\" style=\"display:none\">\n            <div class=\"kq-auth-card\">\n              <h3>Forgot Password<\/h3>\n              <div class=\"kq-field\">\n                <input id=\"kq-forgot-identity\" placeholder=\"Username or Email\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-forgot-newpass\" placeholder=\"New Password\" type=\"password\" \/>\n                <button class=\"kq-toggle-pass\" type=\"button\" aria-label=\"Toggle password\">\ud83d\udc41<\/button>\n              <\/div>\n              <button id=\"kq-forgot-btn\" class=\"button kq-btn-small\">Update Password<\/button>\n            <\/div>\n          <\/div>\n\n        <\/div>\n\n      <\/div>\n    <\/div>\n    \n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<!-- HTML for the Desmos Calculator Embed (Always Visible) -->\n<div id=\"desmos-container\">\n    <iframe loading=\"lazy\"\n        src=\"https:\/\/www.desmos.com\/calculator\/fxgemyy2gl\"\n        width=\"100%\"\n        height=\"500px\"\n        frameborder=\"0\"\n        allowfullscreen\n    ><\/iframe>\n<\/div>\n\n<!-- Button to Open Calculator in Slide-Out Panel -->\n<button id=\"desmos-toggle\" style=\"position: fixed; top: 20px; right: 20px; z-index: 1000;\">\n    Open Calculator\n<\/button>\n\n<!-- Slide-Out Desmos Calculator Panel (hidden initially) -->\n<div id=\"desmos-panel\">\n    <iframe loading=\"lazy\"\n        src=\"https:\/\/www.desmos.com\/calculator\/fxgemyy2gl\"\n        width=\"100%\"\n        height=\"95%\"\n        frameborder=\"0\"\n        allowfullscreen\n    ><\/iframe>\n<\/div>\n\n<!-- CSS Styling for the Slide-Out Panel -->\n<style>\n    \/* Main Container Styling *\/\n    #desmos-container {\n        max-width: 600px; \/* Adjust as needed *\/\n        margin: 20px auto;\n    }\n\n    \/* Slide-Out Panel Styling *\/\n    #desmos-panel {\n        position: fixed;\n        top: 0;\n        right: -400px; \/* Hidden by default *\/\n        width: 400px; \/* Adjust width as needed *\/\n        height: 100vh;\n        background-color: white;\n        border-left: 1px solid #ccc;\n        box-shadow: -2px 0 5px rgba(0, 0, 0, 0.2);\n        transition: right 0.3s ease;\n        z-index: 999; \/* Ensure it overlays content *\/\n    }\n\n    #desmos-panel.open {\n        right: 0;\n    }\n<\/style>\n\n<!-- JavaScript to Toggle the Slide-Out Panel -->\n<script>\n    document.getElementById(\"desmos-toggle\").onclick = function() {\n        var panel = document.getElementById(\"desmos-panel\");\n        if (panel.classList.contains(\"open\")) {\n            panel.classList.remove(\"open\");\n        } else {\n            panel.classList.add(\"open\");\n        }\n    };\n<\/script>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\">Wait for the Desmos Calculator to appear.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">SAT MATH PROBLEM SOLUTIONS WITH STEP-BY-STEP EXPLANATION<\/h3>\n\n\n\n<p>Do not open the tabs before finishing the practice test above! For your convenience, we have compiled all the solutions and their explanations here. We will also give you some tips and advice to help you understand them better. You&#8217;ll see <strong>&#8216;why this answer is correct&#8217;<\/strong> and <strong>&#8216;why this is incorrect.&#8217;<\/strong><\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Math Solutions and Explanations:<\/h4>\n\n\n\n<p>The light red color shows the Question, green shows the Correct answer with step-by-step explanation, red shows the Incorrect one, and blue shows Desmos Tips or Tricks.<\/p>\n\n\n\n<div class=\"wp-block-coblocks-accordion alignfull\">\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>1st Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> If <math data-latex=\"f(x) = \\frac{x^2 - 6x + 3}{x - 1}\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = \\frac{x^2 &#8211; 6x + 3}{x &#8211; 1}<\/annotation><\/semantics><\/math>, what if <math data-latex=\"f(-1)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(-1)<\/annotation><\/semantics><\/math>?<br>A) -5<br>B) -2<br>C) 2<br>D) 5<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 Explain the Question (What is being asked)<\/strong><br>~ You are given a <strong>rational function<\/strong><br>~ You must <strong>substitute x = \u22121<\/strong> into the function<br>~ Since the denominator is <strong>not zero<\/strong> at <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = -1<\/annotation><\/semantics><\/math>, direct substitution is allowed<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Solution<\/strong><br>The questions says:<br><math data-latex=\"f(x) = \\frac{x^2 - 6x + 3}{x - 1}\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = \\frac{x^2 &#8211; 6x + 3}{x &#8211; 1}<\/annotation><\/semantics><\/math><br><br>where <math data-latex=\"f(x) = f(-1)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = f(-1)<\/annotation><\/semantics><\/math><br><br>Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = -1<\/annotation><\/semantics><\/math>: All we need to do is to fill the value <strong>x<\/strong>.<br><br><math data-latex=\"f(-1) = \\frac{(-1)^2 - 6(-1) + 3}{(-1) - 1}\\\\ \\\\f(-1) = \\frac{(-1)^2 - 6(-1) + 3}{-1 - 1}\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow><mrow><mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">f(-1) = \\frac{(-1)^2 &#8211; 6(-1) + 3}{(-1) &#8211; 1}\\\\ \\\\f(-1) = \\frac{(-1)^2 &#8211; 6(-1) + 3}{-1 &#8211; 1}<\/annotation><\/semantics><\/math><br><br>Numerator:<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mn>6<\/mn><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(-1)^2 &#8211; 6(-1) + 3 = 1 + 6 + 3 = 10<\/annotation><\/semantics><\/math><br>Denominator:<math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-1 &#8211; 1 = -2<\/annotation><\/semantics><\/math><br>Final value:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>10<\/mn><mrow><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(-1) = \\frac{10}{-2} = -5<\/annotation><\/semantics><\/math><br><br>\u2705 <strong>Correct Answer: Option A<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>-2<\/strong> \u2192 Divided incorrectly<br>\u274c <strong>2<\/strong> \u2192 Ignored sign of denominator<br>\u274c <strong>5<\/strong> \u2192 Forgot denominator was negative<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>5\ufe0f\u20e3 Desmos ACTUAL Trick<\/strong><br>1. Type: f(x) = (x^2 &#8211; 6x + 3)\/(x -1)<br>2. Type on the 2nd line: f(-1)<br>2. Desmos shows Output: -5<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>2nd Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A veterinarian recommends that each day a certain rabbit should eat 25 calories per pound of the rabbit\u2019s weight, plus an additional 11 calories. Which equation represents this situation, where <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit\u2019s weight is <math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> pounds?<br>A) <math data-latex=\"c = 25x\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>25<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c = 25x<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"c = 25x + 11\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>25<\/mn><mi>x<\/mi><mo>+<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 25x + 11<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"c = 36x\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>36<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c = 36x<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"c = 11x + 25\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 11x + 25<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\u2705 Correct Answer: B) <math><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>25<\/mn><mi>x<\/mi><mo>+<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 25x + 11<\/annotation><\/semantics><\/math><\/strong><br><br><strong>\ud83e\uddee Step-by-Step Correct Reasoning<\/strong><br>The statement has <strong>three parts<\/strong>:<br>1. The weight is <em>x<\/em> and the 25 calory should be given according to weight, so we multiply. <br>2. <strong>25 calories per pound<\/strong> \u2192 <math><semantics><mrow><mn>25<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">25x<\/annotation><\/semantics><\/math> <br>3. <strong>Plus an additional 11 calories<\/strong> \u2192 <math><semantics><mrow><mo>+<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">+ 11<\/annotation><\/semantics><\/math><br>Combine both:<br><math display=\"block\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>25<\/mn><mi>x<\/mi><mo>+<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 25x + 11<\/annotation><\/semantics><\/math><br>\u2714 Matches Option B exactly<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Option A: <math><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>25<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c = 25x<\/annotation><\/semantics><\/math><br><strong>Why students choose it:<\/strong><br>They stop reading after: \u201c25 calories per pound\u201d<br>They ignore: \u201cplus an additional 11 calories\u201d<br>\u274c Missing constant term<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Option C: <math><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>36<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c = 36x<\/annotation><\/semantics><\/math><br><strong>Why students choose it:<\/strong><br>They incorrectly add:<br><math display=\"block\"><semantics><mrow><mn>25<\/mn><mo>+<\/mo><mn>11<\/mn><mo>=<\/mo><mn>36<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">25 + 11 = 36<\/annotation><\/semantics><\/math><br>But 11 is <strong>not per pound<\/strong>, it\u2019s a <strong>fixed amount<\/strong>.<br>\u274c Mixed unit mistake<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Option D: <math><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 11x + 25<\/annotation><\/semantics><\/math><br><strong>Why students choose it:<\/strong><br>They reverse roles:<br>~ Treat 11 as \u201cper pound\u201d<br>~ Treat 25 as constant<br>This directly contradicts the wording.<br>\u274c Misreading variables<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee DESMOS CALCULATOR \u2014 QUICK CHECK<\/strong><br><strong>Step-by-Step<\/strong><br>1. Open <strong>Desmos<\/strong><br>2. Type: c = 25x + 11<br>3. Use the <strong>Table icon<\/strong><br>4. Try <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 1<\/annotation><\/semantics><\/math>:<br>~ Calories = 36<br>~ Matches: 25 + 11<br>\u2714 Confirms equation logic<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>3rd Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><br><math data-latex=\"f(x) = 2(3^x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>3<\/mn><mi>x<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 2(3^x)<\/annotation><\/semantics><\/math><br>For the function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math> defined above, what is the value of <math data-latex=\"f(2)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(2)<\/annotation><\/semantics><\/math>?<br>A) 9<br>B) 12<br>C) 18<br>D) 36<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>1\ufe0f\u20e3 Explain the Question<\/strong><br>~ This is an <strong>exponential function<\/strong><br>~ You substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2<\/annotation><\/semantics><\/math><br>~ Exponents must be evaluated <strong>before multiplication<\/strong><br><br><strong>2\ufe0f\u20e3 Formula \/ Rule Used<\/strong><br><strong>Order of Operations (PEMDAS)<\/strong><br>Exponents \u2192 Multiplication<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Solution<\/strong><br>We know <math data-latex=\"f(x) = f(2)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = f(2)<\/annotation><\/semantics><\/math>, so<br>Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(2) = 2(3^2)<\/annotation><\/semantics><\/math><br>Calculate exponent:<math display=\"block\"><semantics><mrow><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3^2 = 9<\/annotation><\/semantics><\/math><br>Multiply: <math data-latex=\"2(9)\"><semantics><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>9<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2(9)<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>2<\/mn><mo>\u00d7<\/mo><mn>9<\/mn><mo>=<\/mo><mn>18<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2 \\times 9 = 18<\/annotation><\/semantics><\/math><br><math data-latex=\"f(2) = 18\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>18<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(2) = 18<\/annotation><\/semantics><\/math><br>\u2705 <strong>Correct Answer: Option C<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>9<\/strong> \u2192 Forgot to multiply by 2<br>\u274c <strong>12<\/strong> \u2192 Multiplied before exponent<br>\u274c <strong>36<\/strong> \u2192 Squared 6 incorrectly<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>5\ufe0f\u20e3 Desmos ACTUAL Trick<\/strong><br>We know x = 2, so<br>1. Type: 2(3^2)<br>2. Output: 18<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>4th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> For the exponential function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math>, the value of <math data-latex=\"f(0)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(0)<\/annotation><\/semantics><\/math> is <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math>, where <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is a constant. Of the following equations that define the function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math>, which equation shows the value of <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> as the coefficient or the base?<br>A) <math data-latex=\"f(x) = 22(1.5)^{x+1}\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>22<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 22(1.5)^{x+1}<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"f(x) = 33(1.5)^x\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>33<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>x<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 33(1.5)^x<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"f(x) = 49.5(1.5)^{x-1}\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>49.5<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 49.5(1.5)^{x-1}<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"f(x) = 74.25(1.5)^{x-2}\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>74.25<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 74.25(1.5)^{x-2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>1\ufe0f\u20e3 What the Question Is Really Testing<\/strong><br>~ This is about the <strong>y-intercept of an exponential function<\/strong><br>~ The y-intercept occurs when: <math data-latex=\"f(x) = f(0)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = f(0)<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math><br>So:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = c<\/annotation><\/semantics><\/math><br><math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is Constant and coefficient or the base value.<br>The question is asking:<br>In which equation is the value of <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(0)<\/annotation><\/semantics><\/math> <strong>already visible<\/strong> without doing extra exponent work?<br><br><strong>2\ufe0f\u20e3 Key Rule \/ Formula<\/strong><br>For an exponential function:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><msup><mo stretchy=\"false\">)<\/mo><mi>x<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = a(b)^x<\/annotation><\/semantics><\/math><br>When <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>0<\/mn><\/msup><mo>=<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = a(b)^0 = a<\/annotation><\/semantics><\/math><br>\u2705 <strong>The coefficient a equals the y-intercept<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Option Analysis<\/strong><br>The question tells us that the value of <math data-latex=\"f(x) = f(0) = c\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = f(0) = c<\/annotation><\/semantics><\/math><br>It gives us a clean hint that <math data-latex=\"x = 0\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math><br><br>But the most important hint is:<br><math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is Constant and coefficient or <strong>the base value<\/strong>.<br>~ In simple words, <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is the Initial value, so based on formula above: <math data-latex=\"c = a\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c = a<\/annotation><\/semantics><\/math><br>~ <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>: According to the formula, <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> is the base value.<br><br>Overall what it means that the Base value is the Final value, so the solution is:<br><math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> value is the answer.<br>Now look up the Options, which option do you consider stay remained?<br><br><strong>Option B: <\/strong><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>33<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>x<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 33(1.5)^x<\/annotation><\/semantics><\/math><br>To Test &#8211; Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>:<br><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>33<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>0<\/mn><\/msup><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = 33(1.5)^0 = 33<\/annotation><\/semantics><\/math><br>\u2705 <strong>The coefficient directly equals c<\/strong><br>\u2714 No extra calculation needed<br>\u2714 <math data-latex=\"a = c\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a = c<\/annotation><\/semantics><\/math> or Base Value Remains the same at last<br>\u2714 <math data-latex=\"a = 33\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 33<\/annotation><\/semantics><\/math>  and <math data-latex=\"c = 33\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 33<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option A<\/strong><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>22<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 22(1.5)^{x+1}<\/annotation><\/semantics><\/math><br>Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>22<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>1<\/mn><\/msup><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = 22(1.5)^1 = 33<\/annotation><\/semantics><\/math><br>\u274c The value of <math><semantics><mrow><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is <strong>not shown directly<\/strong><br>\u274c <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> are not the same or Base Value doesn&#8217;t remain the same at last.<br>\u274c <math data-latex=\"a = 22\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>22<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 22<\/annotation><\/semantics><\/math>  and  <math data-latex=\"c = 33\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 33<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C<\/strong><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>49.5<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 49.5(1.5)^{x-1}<\/annotation><\/semantics><\/math><br>Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>49.5<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = 49.5(1.5)^{-1}<\/annotation><\/semantics><\/math><br><math data-latex=\"f(0) = 49.5 \\times (\\frac{1}{1.5})^1\\\\ \\\\f(0) = \\frac{49.5}{1.5}\\\\ \\\\f(0) = 33\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>49.5<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mfrac><mn>1<\/mn><mn>1.5<\/mn><\/mfrac><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>1<\/mn><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>49.5<\/mn><mn>1.5<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>33<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">f(0) = 49.5 \\times (\\frac{1}{1.5})^1\\\\ \\\\f(0) = \\frac{49.5}{1.5}\\\\ \\\\f(0) = 33<\/annotation><\/semantics><\/math><br><br>\u274c The value of <math><semantics><mrow><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is <strong>not shown directly<\/strong><br>\u274c <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> are not the same or Base Value doesn&#8217;t remain the same at last.<br>\u274c <math data-latex=\"a = 49.5\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>49.5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 49.5<\/annotation><\/semantics><\/math>  and  <math data-latex=\"c = 33\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 33<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D<\/strong><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>74.25<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 74.25(1.5)^{x-2}<\/annotation><\/semantics><\/math><br>Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>74.25<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mrow><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = 74.25(1.5)^{-2}<\/annotation><\/semantics><\/math><br><math data-latex=\"f(0) = 74.25 \\times (\\frac{1}{1.5})^2\\\\ \\\\f(0) = \\frac{74.25}{2.25}\\\\ \\\\f(0) = 33\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>74.25<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mfrac><mn>1<\/mn><mn>1.5<\/mn><\/mfrac><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>74.25<\/mn><mn>2.25<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>33<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">f(0) = 74.25 \\times (\\frac{1}{1.5})^2\\\\ \\\\f(0) = \\frac{74.25}{2.25}\\\\ \\\\f(0) = 33<\/annotation><\/semantics><\/math><br><br>\u274c The value of <math><semantics><mrow><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> is <strong>not shown directly<\/strong><br>\u274c <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> are not the same or Base Value doesn&#8217;t remain the same at last.<br>\u274c <math data-latex=\"a = 74.25\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>74.25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 74.25<\/annotation><\/semantics><\/math>  and  <math data-latex=\"c = 33\"><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = 33<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong><em>Use Desmos as a calculator for this.<\/em><\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>5th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Which of the following expressions is equivalent to <math data-latex=\"8x^{10} - 8x^9 + 88x\"><semantics><mrow><mn>8<\/mn><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u2212<\/mo><mn>8<\/mn><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>+<\/mo><mn>88<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">8x^{10} &#8211; 8x^9 + 88x<\/annotation><\/semantics><\/math>?<br>A) <math data-latex=\"x(7x^{10} - 7x^9 + 87x)\"><semantics><mrow><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>7<\/mn><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u2212<\/mo><mn>7<\/mn><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>+<\/mo><mn>87<\/mn><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x(7x^{10} &#8211; 7x^9 + 87x)<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"x(8^{10} - 8^9 + 88)\"><semantics><mrow><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>8<\/mn><mn>10<\/mn><\/msup><mo>\u2212<\/mo><msup><mn>8<\/mn><mn>9<\/mn><\/msup><mo>+<\/mo><mn>88<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x(8^{10} &#8211; 8^9 + 88)<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"8x(x^{10} - x^9 + 11x)\"><semantics><mrow><mn>8<\/mn><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">8x(x^{10} &#8211; x^9 + 11x)<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"8x(x^{9} - x^8 + 11)\"><semantics><mrow><mn>8<\/mn><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>8<\/mn><\/msup><mo>+<\/mo><mn>11<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">8x(x^{9} &#8211; x^8 + 11)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 Explain the Question (Interpretation)<\/strong><br>~ You are given a polynomial with <strong>three terms<\/strong><br>~ The question asks for an <strong>equivalent expression<\/strong><br>~ This usually means <strong>factoring<\/strong>, not expanding<br>~ Look for a <strong>common factor<\/strong> across all terms<br><br><strong>2\ufe0f\u20e3 Formula \/ Rule Used<\/strong><br><strong><em>Greatest Common Factor (GCF)<\/em><\/strong><br>If all terms share a factor, factor it out:<br><math display=\"block\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">ab + ac = a(b + c)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Solution<\/strong><br>Given:<math display=\"block\"><semantics><mrow><mn>8<\/mn><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u2212<\/mo><mn>8<\/mn><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>+<\/mo><mn>88<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">8x^{10} &#8211; 8x^9 + 88x<\/annotation><\/semantics><\/math><br><strong>Step 1: Find the GCF<br><\/strong>Coefficients: <strong>8<\/strong>x<sup>10<\/sup> &#8211; <strong>8<\/strong>x<sup>9<\/sup> + <strong>88<\/strong>x<br>GCF of 8, 8, 88 \u2192 <strong>8<\/strong><br>Variables: smallest power of <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> \u2192 <strong>x<\/strong><br>So GCF = <strong>8x<\/strong><br><br><strong>Step 2: Factor out 8x<\/strong><math display=\"block\"><semantics><mrow><mn>8<\/mn><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u00f7<\/mo><mn>8<\/mn><mi>x<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>9<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">8x^{10} \u00f7 8x = x^9<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>8<\/mn><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>\u00f7<\/mo><mn>8<\/mn><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>8<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">-8x^9 \u00f7 8x = -x^8<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>88<\/mn><mi>x<\/mi><mo>\u00f7<\/mo><mn>8<\/mn><mi>x<\/mi><mo>=<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">88x \u00f7 8x = 11<\/annotation><\/semantics><\/math><br>Final factored form:<math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>8<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>8<\/mn><\/msup><mo>+<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{8x(x^9 &#8211; x^8 + 11)}<\/annotation><\/semantics><\/math><br>\u2705 <strong>Correct Answer: Option D<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>Option A:<\/strong><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u2212<\/mo><mn>7<\/mn><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>+<\/mo><mn>87<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x(7x^{10} &#8211; 7x^9 + 87x)<\/annotation><\/semantics><\/math><br>~ Coefficients are changed incorrectly<br>~ <strong>Factoring never changes values<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>Option B:<\/strong><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>8<\/mn><mn>10<\/mn><\/msup><mo>\u2212<\/mo><msup><mn>8<\/mn><mn>9<\/mn><\/msup><mo>+<\/mo><mn>88<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x(8^{10} &#8211; 8^9 + 88)<\/annotation><\/semantics><\/math><br>~ Powers are wrongly applied to <strong>8 instead of x<\/strong><br>~ Completely different expression<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>Option C:<\/strong><math display=\"block\"><semantics><mrow><mn>8<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>10<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>9<\/mn><\/msup><mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">8x(x^{10} &#8211; x^9 + 11x)<\/annotation><\/semantics><\/math><br>~ Incorrect division while factoring<br>~ Inside powers increase instead of decrease<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>5\ufe0f\u20e3 Desmos ACTUAL Trick<\/strong><br>1. Type the question equation: 8x^10 &#8211; 8x^9 + 88x<br>2. Check the graph: You will see line.<br>3. Type Options one-by-one in different lines.<br>4. Observe:<br>~ Both Question and Option D overlaps perfectly. They are on the same track.<br>That confirms our correct option.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>6th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> <br>\ud835\udc66 = 2\ud835\udc65 + 10<br>\ud835\udc66 = 2\ud835\udc65 \u2212 1<br>At how many points do the graphs of the given equations intersect in the xy-plane?<br>A) Zero<br>B) Exactly one<br>C) Exactly two<br>D) Infinitely many<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 Core Concept Used<br><\/strong>Lines intersect <strong>once<\/strong> if slopes are different<br>Lines never intersect if:<br>~ Slopes are the <strong>same<\/strong><br>~ Intercepts are <strong>different<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Solution<\/strong><br>The formula: <math data-latex=\"y = mx + b\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = mx + b<\/annotation><\/semantics><\/math><strong><br>Step 1: Compare slopes<br><\/strong>Both equations have slope:<br><math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = 2<\/annotation><\/semantics><\/math><br>So the lines are <strong>parallel<\/strong>.<br><br><strong>Step 2: Compare y-intercepts<br><\/strong>First line: <math><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = 10<\/annotation><\/semantics><\/math><br>Second line: <math><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = -1<\/annotation><\/semantics><\/math><br>Different intercepts \u2192 different lines<br><br><strong>Step 3: Conclusion<br><\/strong>Parallel lines <strong>never intersect<\/strong>.<br>\u2705 Correct Answer: <strong>Zero<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option B: Exactly one \u274c<\/strong><br><strong>Trap:<\/strong> Student assumes all linear systems intersect once.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C: Exactly two \u274c<\/strong><br><strong>Trap:<\/strong> Student confuses with quadratic intersections.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D: Infinitely many \u274c<\/strong><br><strong>Trap:<\/strong> Would only occur if equations were identical.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Confirmation<br><\/strong>Plot: <br>y = 2x + 10<br>y = 2x &#8211; 1<br>\u2714 Parallel lines, no crossing<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>7th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The expression <math data-latex=\"90y^5 - 54y^4\"><semantics><mrow><mn>90<\/mn><msup><mi>y<\/mi><mn>5<\/mn><\/msup><mo>\u2212<\/mo><mn>54<\/mn><msup><mi>y<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90y^5 &#8211; 54y^4<\/annotation><\/semantics><\/math> is equivalent to <math data-latex=\"ry^4(15y - 9)\"><semantics><mrow><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">ry^4(15y &#8211; 9)<\/annotation><\/semantics><\/math>, where <math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math> is a constant. What is the value of <math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>?<br>A) 6<br>B) 9<br>C) 15<br>D) 90<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 What the Question Is Testing<\/strong><br>~ This is <strong>factoring<\/strong><br>~ You must <strong>match coefficients<\/strong><br>~ The structure:<math display=\"block\"><semantics><mrow><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">ry^4(15y &#8211; 9)<\/annotation><\/semantics><\/math><br>~ <math data-latex=\"90y^5 - 54y^4\\ =\\ ry^4(15y - 9)\"><semantics><mrow><mn>90<\/mn><msup><mi>y<\/mi><mn>5<\/mn><\/msup><mo>\u2212<\/mo><mn>54<\/mn><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mtext>&nbsp;<\/mtext><mo>=<\/mo><mtext>&nbsp;<\/mtext><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">90y^5 &#8211; 54y^4\\ =\\ ry^4(15y &#8211; 9)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Factoring<\/strong><br>Start with:<math display=\"block\"><semantics><mrow><mn>90<\/mn><msup><mi>y<\/mi><mn>5<\/mn><\/msup><mo>\u2212<\/mo><mn>54<\/mn><msup><mi>y<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">90y^5 &#8211; 54y^4<\/annotation><\/semantics><\/math><br>Factor out the <strong>greatest common factor<\/strong>:<br>~ GCF of 90 and 54 \u2192 <strong>18<\/strong><br>~ Lowest power of <math><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math> \u2192 <math><semantics><mrow><msup><mi>y<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">y^4<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>90<\/mn><msup><mi>y<\/mi><mn>5<\/mn><\/msup><mo>\u2212<\/mo><mn>54<\/mn><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo>=<\/mo><mn>18<\/mn><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">90y^5 &#8211; 54y^4 = 18y^4(5y &#8211; 3)<\/annotation><\/semantics><\/math><br>Now rewrite the given expression:<math display=\"block\"><semantics><mrow><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">ry^4(15y &#8211; 9)<\/annotation><\/semantics><\/math><br>Factor inside:<math display=\"block\"><semantics><mrow><mn>15<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">15y &#8211; 9 = 3(5y &#8211; 3)<\/annotation><\/semantics><\/math><br>So:<math display=\"block\"><semantics><mrow><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>3<\/mn><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">ry^4(15y &#8211; 9) = 3r y^4(5y &#8211; 3)<\/annotation><\/semantics><\/math><br><br><strong>3\ufe0f\u20e3 Match the Coefficients<\/strong><br>From factoring:<math display=\"block\"><semantics><mrow><mn>18<\/mn><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">18y^4(5y &#8211; 3)<\/annotation><\/semantics><\/math><br>From rewritten form:<math display=\"block\"><semantics><mrow><mn>3<\/mn><mi>r<\/mi><msup><mi>y<\/mi><mn>4<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>y<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3r y^4(5y &#8211; 3)<\/annotation><\/semantics><\/math><br>So:<math display=\"block\"><semantics><mrow><mn>3<\/mn><mi>r<\/mi><mo>=<\/mo><mn>18<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3r = 18<\/annotation><\/semantics><\/math><br>Solve:<math display=\"block\"><semantics><mrow><mi>r<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r = 6<\/annotation><\/semantics><\/math><br><strong>\u2705 Correct Answer: Option A<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\ud83e\uddee <strong>Desmos Trick<\/strong><br>1. Type: x = 90y^5 &#8211; 54y^4<br>2. Do not directly type: ry^4(15y &#8211; 9)<br>3. What we need to do is to put value of <math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math>:<br>~ The options are values of <math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><br>~ So replace it with 6, 9, 15, 90.<br>~ Type in 2nd line like this: x = 6y^4(15y &#8211; 9)<br>4. Observe:<br>~ The graph lines overlap &#8211; confirms correctness of the Option.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>8th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A rectangular volleyball court has an area of 162 square meters. If the length of the court is twice the width, what is the width of the court, in meters?<br>A) 9<br>B) 18<br>C) 27<br>D) 54<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 Explain the Question<\/strong><br>~ Rectangle area is given<br>~ Relationship between length and width is given<br>~ We are solving for <strong>width<\/strong><br>~ This is a <strong>systems \/ substitution<\/strong> word problem<br><br><strong>2\ufe0f\u20e3 Formula \/ Rule Used<\/strong><br><strong>Area of a Rectangle<\/strong><math display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mi>l<\/mi><mo>\u00d7<\/mo><mi>w<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A = l \\times w<\/annotation><\/semantics><\/math><br>Area of Rectangle: <math data-latex=\"A\"><semantics><mi>A<\/mi><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><br>Length = <math data-latex=\"l\"><semantics><mi>l<\/mi><annotation encoding=\"application\/x-tex\">l<\/annotation><\/semantics><\/math><br>Width = <math data-latex=\"w\"><semantics><mi>w<\/mi><annotation encoding=\"application\/x-tex\">w<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Solution<\/strong><br>Let:<br>~ Width = <math><semantics><mrow><mi>w<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">w<\/annotation><\/semantics><\/math><br>~ Length = <math><semantics><mrow><mn>2<\/mn><mi>w<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2w<\/annotation><\/semantics><\/math> (Question told: If the <strong>length is<\/strong> <strong>twice<\/strong> the width)<br>Substitute into area formula: <math data-latex=\"A = l \\times w\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mi>l<\/mi><mo>\u00d7<\/mo><mi>w<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A = l \\times w<\/annotation><\/semantics><\/math><br>We know that <math data-latex=\"l = 2w\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mn>2<\/mn><mi>w<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = 2w<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mn>162<\/mn><mo>=<\/mo><mn>2<\/mn><mi>w<\/mi><mo>\u22c5<\/mo><mi>w<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">162 = 2w \\cdot w<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>162<\/mn><mo>=<\/mo><mn>2<\/mn><msup><mi>w<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">162 = 2w^2<\/annotation><\/semantics><\/math><br>Divide by 2 both sides or just move 2 to left-hand side in divide position:<br><math display=\"block\"><semantics><mrow><msup><mi>w<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>81<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">w^2 = 81<\/annotation><\/semantics><\/math><br>Take square root: <math data-latex=\"w = \\sqrt{81}\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><msqrt><mn>81<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">w = \\sqrt{81}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">w = 9<\/annotation><\/semantics><\/math><br>\u2705 <strong>Correct Answer: Option A<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>18:<\/strong> Student forgot to square root<br>\u274c <strong>27:<\/strong> Multiplied instead of solving<br>\u274c <strong>54:<\/strong> Treated area as perimeter<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>5\ufe0f\u20e3 Desmos ACTUAL Trick<\/strong><br>1. Type in Desmos: 2w^2 = 162<br>2. Desmos solves: w = 9<br>But if you want to see visually in graph then<br>1. Type: y = 2x^2<br>2. Type in 2nd line: y = 162<br>3. Observe: The graph Intersection at x = 9.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>9th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> In right triangle <math data-latex=\"RST\"><semantics><mrow><mi>R<\/mi><mi>S<\/mi><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">RST<\/annotation><\/semantics><\/math>, the sum of the measures of angle <math data-latex=\"R\"><semantics><mi>R<\/mi><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math> and angle <math data-latex=\"S\"><semantics><mi>S<\/mi><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math> is 90 degrees. The value of <math data-latex=\"sin(R)\"><semantics><mrow><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">sin(R)<\/annotation><\/semantics><\/math> is <math data-latex=\"\\frac{\\sqrt{15}}{4}\"><semantics><mfrac><msqrt><mn>15<\/mn><\/msqrt><mn>4<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\sqrt{15}}{4}<\/annotation><\/semantics><\/math>. What is the value of <math data-latex=\"cos(S)\"><semantics><mrow><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">cos(S)<\/annotation><\/semantics><\/math>?<br><br>A) <math data-latex=\"\\frac{\\sqrt{15}}{15}\"><semantics><mfrac><msqrt><mn>15<\/mn><\/msqrt><mn>15<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\sqrt{15}}{15}<\/annotation><\/semantics><\/math><br><br>B) <math data-latex=\"\\frac{\\sqrt{15}}{4}\"><semantics><mfrac><msqrt><mn>15<\/mn><\/msqrt><mn>4<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\sqrt{15}}{4}<\/annotation><\/semantics><\/math><br><br>C) <math data-latex=\"\\frac{4\\sqrt{15}}{15}\"><semantics><mfrac><mrow><mn>4<\/mn><msqrt><mn>15<\/mn><\/msqrt><\/mrow><mn>15<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{4\\sqrt{15}}{15}<\/annotation><\/semantics><\/math><br><br>D) <math data-latex=\"\\sqrt{15}\"><semantics><msqrt><mn>15<\/mn><\/msqrt><annotation encoding=\"application\/x-tex\">\\sqrt{15}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\"><strong>\u2705 Understand the QUESTION<br><\/strong>Trigonometry: Complementary Angles<br><strong>Given:<\/strong><br>~ Triangle RST is a <strong>right triangle<\/strong><br><math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>R<\/mi><mo>+<\/mo><mi mathvariant=\"normal\">\u2220<\/mi><mi>S<\/mi><mo>=<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle R + \\angle S = 90^\\circ<\/annotation><\/semantics><\/math><br>That means <strong>R and S are complementary angles<\/strong><br><br><math><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msqrt><mn>15<\/mn><\/msqrt><mn>4<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">\\sin(R) = \\dfrac{\\sqrt{15}}{4}<\/annotation><\/semantics><\/math><br><strong>Asked:<\/strong><br>Find <math><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos(S)<\/annotation><\/semantics><\/math><br><br><strong>\ud83e\udde0 Key Trigonometry Rule<\/strong><br>In a right triangle:<math display=\"block\"><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin(\\theta) = \\cos(90^\\circ &#8211; \\theta)<\/annotation><\/semantics><\/math><br>Since:<math display=\"block\"><semantics><mrow><mi>S<\/mi><mo>=<\/mo><msup><mn>90<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S = 90^\\circ &#8211; R<\/annotation><\/semantics><\/math><br><strong><em>Go back to the definitions (this is the real \u201cwhy\u201d)<br><\/em><\/strong>In a <strong>right triangle<\/strong>:<math display=\"block\"><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mtext>opposite<\/mtext><mtext>hypotenuse<\/mtext><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sin(\\theta) = \\frac{\\text{opposite}}{\\text{hypotenuse}}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mtext>adjacent<\/mtext><mtext>hypotenuse<\/mtext><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos(\\theta) = \\frac{\\text{adjacent}}{\\text{hypotenuse}}<\/annotation><\/semantics><\/math><br>Now look at angles <strong>R and S<\/strong>:<br>What is <strong>opposite<\/strong> to R \u2192 is <strong>adjacent<\/strong> to S<br>What is <strong>adjacent<\/strong> to R \u2192 is <strong>opposite<\/strong> to S<br>So the <strong>same two sides<\/strong> are used \u2014 just swapped roles.<br>That\u2019s why:<math display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>R<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos(S) = \\sin(R)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step 1: Core Trigonometric Rule Used<br><\/strong>For complementary angles in a right triangle:<br>~ Let&#8217;s understand why <math data-latex=\"cos(S) = sin(R)\"><semantics><mrow><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">cos(S) = sin(R)<\/annotation><\/semantics><\/math> are the same.<br>~ The question tells us: <math data-latex=\"\\angle R + \\angle S = 90 \\degree\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>S<\/mi><mo>=<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle R + \\angle S = 90 \\degree<\/annotation><\/semantics><\/math><br>~ That means <math data-latex=\"\\angle R = 90 \\degree - \\angle S\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mo>=<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>\u2220<\/mi><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle R = 90 \\degree &#8211; \\angle S<\/annotation><\/semantics><\/math>, and <math data-latex=\"\\angle S = 90 \\degree - \\angle R\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mo>=<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>\u2220<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle S = 90 \\degree &#8211; \\angle R<\/annotation><\/semantics><\/math><br>So: <math data-latex=\"sin(R) = cos(90\\degree - R)\"><semantics><mrow><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">sin(R) = cos(90\\degree &#8211; R)<\/annotation><\/semantics><\/math> and <math data-latex=\"cos(S) = sin(90\\degree - S)\"><semantics><mrow><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">cos(S) = sin(90\\degree &#8211; S)<\/annotation><\/semantics><\/math><br><strong><em>That is why:<br><\/em><\/strong><math data-latex=\"sin(R) = cos(S)\"><semantics><mrow><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">sin(R) = cos(S)<\/annotation><\/semantics><\/math>  <strong>\u2192<\/strong>  <math data-latex=\"sin(R) = cos(90\\degree - R)\"><semantics><mrow><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">sin(R) = cos(90\\degree &#8211; R)<\/annotation><\/semantics><\/math>,<br>so <math data-latex=\"cos(S) = cos(90\\degree - R)\"><semantics><mrow><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">cos(S) = cos(90\\degree &#8211; R)<\/annotation><\/semantics><\/math>  \u2192  <math data-latex=\"S = 90\\degree - R\"><semantics><mrow><mi>S<\/mi><mo>=<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S = 90\\degree &#8211; R<\/annotation><\/semantics><\/math><br><br>Do you get is now? For knowledge, you can check <strong>sin(S)<\/strong> too.<br><math data-latex=\"cos(S) = sin(R)\"><semantics><mrow><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">cos(S) = sin(R)<\/annotation><\/semantics><\/math>  \u2192  <math data-latex=\"cos(S) = sin(90\\degree - S)\"><semantics><mrow><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">cos(S) = sin(90\\degree &#8211; S)<\/annotation><\/semantics><\/math>,<br>so <math data-latex=\"sin(R) = sin(90\\degree - S)\"><semantics><mrow><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>S<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">sin(R) = sin(90\\degree &#8211; S)<\/annotation><\/semantics><\/math>  <strong>\u2192<\/strong>  <math data-latex=\"R = 90\\degree - S\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = 90\\degree &#8211; S<\/annotation><\/semantics><\/math><br><br>We know now that <math data-latex=\"S = 90\\degree - R\"><semantics><mrow><mi>S<\/mi><mo>=<\/mo><mn>90<\/mn><mi>\u00b0<\/mi><mo>\u2212<\/mo><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S = 90\\degree &#8211; R<\/annotation><\/semantics><\/math><br><br><strong>\u270f\ufe0f Apply the Rule to Understand<\/strong><br>Look above &#8220;Go back to definition&#8221; part to better understand, why they are the same.<br><math display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mi>S<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><msqrt><mn>15<\/mn><\/msqrt><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos(S) = \\frac{\\sqrt{15}}{4}<\/annotation><\/semantics><\/math><br>Direct complementary-angle identity.<br><strong>Option B is correct.<\/strong><br><strong><em>If you are still not satisfied, take a Practical Visual Desmos test to understand better.<\/em><\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option A:<br><\/strong>Wrong ratio; unnecessary simplification.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C:<br><\/strong>Reciprocal mistake.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D:<br><\/strong>Trigonometric values must be \u2264 1.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83d\udccc Concept Reminder<br>Sine of one acute angle equals cosine of the other<\/strong> in right triangles.<br><br><strong>DESMOS Tricks<br><\/strong>1. Type in one line: sin(x)<br>2. Type in second line: cos(90-x)<br>3. Observe:<br>~ The graphs overlap exactly<br>~ This visually proves: sin(R) = cos(S)<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>10th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The pressure exerted on a scuba diver at sea level is 14.70 pounds per square inch (psi). For each foot the scuba diver descends below sea level, the pressure exerted on the scuba diver increases by 0.44 psi. What is the total pressure, in psi, exerted on the scuba diver at 105 feet below sea level?<br>A) 60.90<br>B) 31.50<br>C) 14.70<br>D) 0.44<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 Core Concept (Why this is a linear model)<\/strong><br>This situation has:<br>~ a <strong>starting value<\/strong> (pressure at sea level)<br>~ a <strong>constant rate of increase per foot<\/strong><br>So we model it using:<br><math display=\"block\"><semantics><mrow><mtext>Total&nbsp;pressure<\/mtext><mo>=<\/mo><mtext>initial&nbsp;pressure<\/mtext><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mtext>rate<\/mtext><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mtext>depth<\/mtext><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Total pressure} = \\text{initial pressure} + (\\text{rate})(\\text{depth})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct.<br><strong>\ud83e\uddee Step-by-Step Solution<\/strong><br><strong>Step 1: Identify known values<\/strong><br>~ Initial pressure = <strong>14.70 psi<\/strong><br>~ Rate of increase = <strong>0.44 psi per foot<\/strong><br>~ Depth = <strong>105 feet<\/strong><br><br><strong>Step 2: Calculate pressure increase due to depth<br><\/strong>This step is necessary because the increase happens <strong>every foot<\/strong>, not once: (Rate) <math data-latex=\"\\times\"><semantics><mo lspace=\"0em\" rspace=\"0em\">\u00d7<\/mo><annotation encoding=\"application\/x-tex\">\\times<\/annotation><\/semantics><\/math> (Depth)<br><math display=\"block\"><semantics><mrow><mn>0.44<\/mn><mo>\u00d7<\/mo><mn>105<\/mn><mo>=<\/mo><mn>46.20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0.44 \\times 105 = 46.20<\/annotation><\/semantics><\/math><br>This is the <strong>additional pressure<\/strong> caused by going down 105 feet.<br><br><strong>Step 3: Add sea-level pressure (Initial)<\/strong><br>The diver already experiences pressure at sea level, so we <strong>must add it<\/strong>.<br><math display=\"block\"><semantics><mrow><mn>14.70<\/mn><mo>+<\/mo><mn>46.20<\/mn><mo>=<\/mo><mn>60.90<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">14.70 + 46.20 = 60.90<\/annotation><\/semantics><\/math><br>\u2705 Correct Answer: <strong>A) 60.90<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option B: 31.50 \u274c<\/strong><br><strong>Trap:<\/strong> Student multiplies incorrectly or uses half the depth.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C: 14.70 \u274c<\/strong><br><strong>Trap:<\/strong> Student ignores depth entirely.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D: 0.44 \u274c<\/strong><br><strong>Trap:<\/strong> Student mistakes rate for total pressure.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Confirmation<br><\/strong>Type: 14.7 + 0.44(105)<br>\u2714 Output = <strong>60.9<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>11th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><br><math data-latex=\"s(n) = 38,000a^n\"><semantics><mrow><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>38,000<\/mn><msup><mi>a<\/mi><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">s(n) = 38,000a^n<\/annotation><\/semantics><\/math><br>The function <math data-latex=\"S\"><semantics><mi>S<\/mi><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math> above models the annual salary, in dollars, of an employee <math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> years after starting a job, where <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> is a constant. If the employee\u2019s salary increases by 4% each year, what is the value of <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>?<br>A) 0.04<br>B) 0.4<br>C) 1.04<br>D) 1.4<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>1\ufe0f\u20e3 Explain the Question<\/strong><br>~ This is an <strong>exponential growth model<\/strong><br>~ <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> represents the <strong>growth factor<\/strong><br>~ A percent increase must be converted correctly<br><br><strong>2\ufe0f\u20e3 Formula \/ Rule Used<\/strong><br>The main formula: <math data-latex=\"s(n) = I(1 + \\frac{4}{100})^n\"><semantics><mrow><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>I<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mn>4<\/mn><mn>100<\/mn><\/mfrac><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>n<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">s(n) = I(1 + \\frac{4}{100})^n<\/annotation><\/semantics><\/math><br><br>But the question only asked to find growth, so: <math data-latex=\"(1 + \\frac{4}{100})\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mn>4<\/mn><mn>100<\/mn><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1 + \\frac{4}{100})<\/annotation><\/semantics><\/math><br><strong>Growth Factor Rule<\/strong><br>If something increases by <math><semantics><mrow><mi>r<\/mi><mi mathvariant=\"normal\">%<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r\\%<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mtext>Growth&nbsp;factor<\/mtext><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Growth factor} = 1 + r<\/annotation><\/semantics><\/math><br><math data-latex=\"r\"><semantics><mi>r<\/mi><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math> is 4% or <math data-latex=\"\\frac{4}{100}\"><semantics><mfrac><mn>4<\/mn><mn>100<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{4}{100}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Solution<\/strong><br>Given:<br>~ Increase = <strong>4%<\/strong><br>~ Convert percent to decimal:<math display=\"block\"><semantics><mrow><mn>4<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>0.04<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4\\% = 0.04<\/annotation><\/semantics><\/math><br>Apply rule:<math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mn>0.04<\/mn><mo>=<\/mo><mn>1.04<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 1 + 0.04 = 1.04<\/annotation><\/semantics><\/math><br>\u2705 <strong>Correct Answer: Option C<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>0.04:<\/strong> Student forgot to add 1<br>\u274c <strong>0.4:<\/strong> Miscalculated percentage and forgot to add 1<br>\u274c <strong>1.4:<\/strong> Miscalculated percentage<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Use DESMOS as a calculator for this.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>12th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Which of the following is an equivalent form of <math data-latex=\"(1.5x - 2.4)^2 - (5.2x^2 - 6.4)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1.5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2.4<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6.4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1.5x &#8211; 2.4)^2 &#8211; (5.2x^2 &#8211; 6.4)<\/annotation><\/semantics><\/math>?<br>A) <math data-latex=\"-2.2x^2 + 1.6\"><semantics><mrow><mo>\u2212<\/mo><mn>2.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>1.6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-2.2x^2 + 1.6<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"-2.2x^2 + 11.2\"><semantics><mrow><mo>\u2212<\/mo><mn>2.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>11.2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-2.2x^2 + 11.2<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"-2.95x^2 - 7.2x + 12.16\"><semantics><mrow><mo>\u2212<\/mo><mn>2.95<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12.16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-2.95x^2 &#8211; 7.2x + 12.16<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"-2.95x^2 - 7.2x + 0.64\"><semantics><mrow><mo>\u2212<\/mo><mn>2.95<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>0.64<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-2.95x^2 &#8211; 7.2x + 0.64<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 Key Algebra Rules Used<\/strong><br><strong>Square a binomial<\/strong><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(a &#8211; b)^2 = a^2 &#8211; 2ab + b^2<\/annotation><\/semantics><\/math><br>~ <strong>Distribute negative sign<\/strong><br>~ <strong>Combine like terms<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Expansion<\/strong><br>Step 1: Square the binomial<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2.4<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(1.5x &#8211; 2.4)^2<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><mi>x<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>1.5<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2.4<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>2.4<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">= (1.5x)^2 &#8211; 2(1.5x)(2.4) + (2.4)^2<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>=<\/mo><mn>2.25<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5.76<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">= 2.25x^2 &#8211; 7.2x + 5.76<\/annotation><\/semantics><\/math><br>Step 2: Subtract the second expression<math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>5.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6.4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>5.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>6.4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">&#8211; (5.2x^2 &#8211; 6.4) = -5.2x^2 + 6.4<\/annotation><\/semantics><\/math><br>Step 3: Combine like terms:<br><math data-latex=\"2.25x^2 - 7.2x + 5.76 - 5.2x^2 + 6.4\\\\ \\\\ 2.25x^2 - 5.2x^2 -7.2x + 5.76 + 6.4\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mn>2.25<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5.76<\/mn><mo>\u2212<\/mo><mn>5.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>6.4<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>2.25<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>5.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5.76<\/mn><mo>+<\/mo><mn>6.4<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">2.25x^2 &#8211; 7.2x + 5.76 &#8211; 5.2x^2 + 6.4\\\\ \\\\ 2.25x^2 &#8211; 5.2x^2 -7.2x + 5.76 + 6.4<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mn>2.25<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>5.2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo>\u2212<\/mo><mn>2.95<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">2.25x^2 &#8211; 5.2x^2 = -2.95x^2<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mtext>&nbsp;stays<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">-7.2x \\text{ stays}<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>5.76<\/mn><mo>+<\/mo><mn>6.4<\/mn><mo>=<\/mo><mn>12.16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5.76 + 6.4 = 12.16<\/annotation><\/semantics><\/math><br>Final expression:<math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>2.95<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>7.2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12.16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-2.95x^2 &#8211; 7.2x + 12.16<\/annotation><\/semantics><\/math><br><strong>\u2705 Correct Answer: Option C<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\ud83e\uddee <strong>Desmos Trick<\/strong><br>1. Type: y = (1.5x &#8211; 2.4)^2 &#8211; (5.2x^2 &#8211; 6.4)<br>2. Type from 2nd line all options one-by-one: y = -2.95x^2 &#8211; 7.2x + 12.16<br>3. Observe:<br>~ The graph lines overlap &#8211; confirms correctness of the Option.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>13th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Square X has a side length of 12 centimeters. The perimeter of square Y is 2 times the perimeter of square X. What is the length, in centimeters, of one side of square Y?<br>A) 6<br>B) 10<br>C) 14<br>D) 24<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>\u2705 Understand the QUESTION<br><\/strong>Perimeter of Squares<br><strong>Given:<\/strong><br>~ Square X has side length = <strong>12 cm<\/strong><br>~ Perimeter of square Y, <em>2 times X<\/em> = <strong>2 \u00d7 perimeter of square X<\/strong><br><strong>Asked:<\/strong><br>Side length of square Y<br><br><strong>\ud83e\udde0 Key Geometry Formula<\/strong><math display=\"block\"><semantics><mrow><mtext>Perimeter&nbsp;of&nbsp;square<\/mtext><mo>=<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mtext>side<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Perimeter of square} = 4 \\times \\text{side}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\"><strong>\u270f\ufe0f Step-by-Step Solution<\/strong><br>Step 1: Perimeter of square X: 4 \u00d7 side<br>4 \u00d7 12 = 48<br>Step 2: Perimeter of square Y:  2 \u00d7 Square X<br>2 \u00d7 48 = 96<br>Step 3: Side of square Y:<br>A Square has 4 sides, so we divide it by 4 to get one side.<br>96 \u00f7 4 = 24<br><strong>\u2705 Correct Answer: Option D<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>6 \u274c<\/strong><br>Directly divide 12 by 2, don&#8217;t do it.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>10 \u274c<\/strong><br>Directly subtract 2 from 12.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>14 \u274c<\/strong><br>Directly add 2 to 12.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>DESMOS Tricks<br><\/strong>It is better to just use Desmos as a calculator for this one.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>14th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math> is defined by <math data-latex=\"f(x) = 7x - 84\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>84<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 7x &#8211; 84<\/annotation><\/semantics><\/math>. What is the <em>x<\/em>-intercept of the graph of <math data-latex=\"y = f(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = f(x)<\/annotation><\/semantics><\/math> in the <em>xy<\/em>-plane?<br>A) (-12, 0)<br>B) (-7, 0)<br>C) (7, 0)<br>D) (12, 0)<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 Core Concept (Why x-intercept means this)<br><\/strong>The <strong>x-intercept<\/strong> is the point where: Take reference from options (<em>x, y<\/em>) <br><math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 0<\/annotation><\/semantics><\/math><br>So we must <strong>set the function equal to zero<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Solution<br>Step 1: Set <\/strong><math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 0<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>84<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">7x &#8211; 84 = 0<\/annotation><\/semantics><\/math><br><strong>Step 2: Solve for <\/strong><math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><br>Move add to right-hand side, which changes its sign from negative to positive:<br><math display=\"block\"><semantics><mrow><mn>7<\/mn><mi>x<\/mi><mo>=<\/mo><mn>84<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">7x = 84<\/annotation><\/semantics><\/math><br>Divide by 7:<br><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 12<\/annotation><\/semantics><\/math><br><strong>Step 3: Write as an ordered pair<br><\/strong><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>12<\/mn><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(12, 0)<\/annotation><\/semantics><\/math><br>\u2705 Correct Answer: <strong>D) (12, 0)<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option A: (-12, 0) \u274c<\/strong><br><strong>Trap:<\/strong> Student moves 84 to the wrong side and changes sign incorrectly.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option B: (-7, 0) \u274c<\/strong><br><strong>Trap:<\/strong> Student divides 84 by 12 instead of 7.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C: (7, 0) \u274c<\/strong><br><strong>Trap:<\/strong> Student confuses coefficient with solution.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Confirmation<br><\/strong>1. Type: y = 7x &#8211; 84<br>2. Look the graph<br>~ Graph crosses x-axis at <strong>x = 12<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>15th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2026\/01\/image_2026-01-13_213810657.png\" alt=\"Learn to solve graphs and improve your math skills to crack in competitive exams\" class=\"wp-image-8767\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The box plots summarize the masses, in kilograms, of two groups of gazelles. Based on the box plots, which of the following statements must be true?<br>A) The mean mass of group 1 is greater than the mean mass of group 2.<br>B) The mean mass of group 1 is less than the mean mass of group 2.<br>C) The median mass of group 1 is greater than the median mass of group 2.<br>D) The median mass of group 1 is less than the median mass of group 2.<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 What This Question Is Testing<\/strong><br>This is <strong>SAT Data Analysis<\/strong>.<br>Key idea:<br><strong>Box plots show medians and quartiles, NOT means.<\/strong><br>So:<br>~ You <strong>can compare medians<\/strong><br>~ You <strong>cannot determine means<\/strong> from a box plot alone<br><br><strong>2\ufe0f\u20e3 Critical Rules You Must Know<\/strong><br>\ud83d\udccc Box Plot Facts<br><strong>Median<\/strong> \u2192 vertical line inside the box<br><strong>Mean<\/strong> \u2192 NOT shown<br>Statements about means <strong>cannot be guaranteed<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Read the Box Plots Carefully<\/strong><br>From the image:<br>~ Median of <strong>Group 1<\/strong> is <strong>to the right<\/strong> of the median of <strong>Group 2<\/strong><br>~ That means: <math display=\"block\"><semantics><mrow><mtext>Median(Group&nbsp;1)<\/mtext><mo>&gt;<\/mo><mtext>Median(Group&nbsp;2)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Median(Group 1)} &gt; \\text{Median(Group 2)}<\/annotation><\/semantics><\/math><br>4\ufe0f\u20e3 Correct Answer Explained<br>\u2705 <strong>C) The median mass of group 1 is greater than the median mass of group 2<\/strong><br>This is <strong>directly visible<\/strong> from the median lines.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">5\ufe0f\u20e3 Why the Other Options Are Wrong (REAL SAT TRAPS)<br>\u274c <strong>A) Mean of group 1 &gt; mean of group 2<\/strong><br>\u274c <strong>B) Mean of group 1 &lt; mean of group 2<\/strong><br>\u2192 <strong>Means are not shown<\/strong><br>\u2192 Distribution shape can change the mean<br>\u2192 SAT trick: <strong>never assume mean from a box plot<\/strong><br>\u274c <strong>D) Median of group 1 &lt; median of group 2<\/strong><br>\u2192 Opposite of what the graph clearly shows<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Note (Why It\u2019s Not Used Here)<br><\/strong>Desmos <strong>cannot compute medians from box plots<\/strong> unless raw data is given.<br>This question is <strong>visual reasoning<\/strong>, not computation.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>16th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math> is defined by <math data-latex=\"f(x) = a(2.2^x + 2.2^b)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>x<\/mi><\/msup><mo>+<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = a(2.2^x + 2.2^b)<\/annotation><\/semantics><\/math>, where <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are integer constants and &lt;math data-latex=&quot;0 &lt; a <b><semantics><mrow><mn>0<\/mn><mo>&lt;<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">0 &lt; a &lt; b<\/annotation><\/semantics><\/math>. The functions <math data-latex=\"g\"><semantics><mi>g<\/mi><annotation encoding=\"application\/x-tex\">g<\/annotation><\/semantics><\/math> and <math data-latex=\"h\"><semantics><mi>h<\/mi><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math> are equivalent to function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math>, where <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> and <math data-latex=\"m\"><semantics><mi>m<\/mi><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math> are constants. Which of the following equations displays the <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-coordinate of the <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-intercept of the graph of <math data-latex=\"y = f(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = f(x)<\/annotation><\/semantics><\/math> in the <math data-latex=\"xy\"><semantics><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">xy<\/annotation><\/semantics><\/math>-plane as a constant or coefficient?<br>I. <math data-latex=\"g(x) = a(2.2^x + k)\"><semantics><mrow><mi>g<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>x<\/mi><\/msup><mo>+<\/mo><mi>k<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">g(x) = a(2.2^x + k)<\/annotation><\/semantics><\/math><br>II. <math data-latex=\"h(x) = a(2.2)^x + m\"><semantics><mrow><mi>h<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2.2<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>x<\/mi><\/msup><mo>+<\/mo><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h(x) = a(2.2)^x + m<\/annotation><\/semantics><\/math><br>A) I only<br>B) II only<br>C) I and II<br>D) Neither I nor II<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>Understand the Question<br><\/strong>The function: <math data-latex=\"f(x) = a(2.2^x + 2.2^b)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>x<\/mi><\/msup><mo>+<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = a(2.2^x + 2.2^b)<\/annotation><\/semantics><\/math><br>~ where <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are <strong><em>constant<\/em><\/strong> integer<br>~ and &lt;math data-latex=&quot;0 &lt; a <b><semantics><mrow><mn>0<\/mn><mo>&lt;<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">0 &lt; a &lt; b<\/annotation><\/semantics><\/math> (<math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> is bigger than value 0 and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> is bigger than <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>)<br><br>The functions <math data-latex=\"g\"><semantics><mi>g<\/mi><annotation encoding=\"application\/x-tex\">g<\/annotation><\/semantics><\/math> and <math data-latex=\"h\"><semantics><mi>h<\/mi><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math> are equivalent to function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math><br>~ where <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> and <math data-latex=\"m\"><semantics><mi>m<\/mi><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math> are <strong><em>constants<\/em><\/strong> <br>it means <math data-latex=\"k = 2.2^b\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">k = 2.2^b<\/annotation><\/semantics><\/math> and <math data-latex=\"m = 2.2^b\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">m = 2.2^b<\/annotation><\/semantics><\/math><br><br>The graph of <math data-latex=\"y = f(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = f(x)<\/annotation><\/semantics><\/math> in the <math data-latex=\"xy\"><semantics><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">xy<\/annotation><\/semantics><\/math>-plane<br><br>I. <math data-latex=\"g(x) = a(2.2^x + k)\"><semantics><mrow><mi>g<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>x<\/mi><\/msup><mo>+<\/mo><mi>k<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">g(x) = a(2.2^x + k)<\/annotation><\/semantics><\/math><br>II. <math data-latex=\"h(x) = a(2.2)^x + m\"><semantics><mrow><mi>h<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2.2<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>x<\/mi><\/msup><mo>+<\/mo><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h(x) = a(2.2)^x + m<\/annotation><\/semantics><\/math><br>Which equation <strong><em>displays<\/em><\/strong> the <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-coordinate of <strong>the <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-intercept<\/strong> as a constant or coefficient?<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 STEP 1: What the question is REALLY testing (critical)<\/strong><br>The question is <strong>not<\/strong> asking:<br>~ which form is equivalent or<br>~ which form gives the y-intercept when substituted<br>It is asking something <strong>much stricter<\/strong>:<br>~ Does the equation <strong>visibly display<\/strong> the y-intercept <strong>as a constant or coefficient<\/strong>?<br>That means:<br>~ The y-intercept must be <strong>directly readable<\/strong><br>~ Not hidden inside a parameter<br>~ Not requiring substitution or calculation<br>This distinction is where the trap is.<br><br><strong>\ud83e\udde0 STEP 2: Find the actual y-intercept of <\/strong><math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math><br>y-intercept occurs where:<math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math><br>Substitute into <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mn>0<\/mn><\/msup><mo>+<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = a(2.2^0 + 2.2^b)<\/annotation><\/semantics><\/math><br>The exponent <strong>0<\/strong> means <strong>1<\/strong>:<br><math data-latex=\"2.2^2 = 1 \\times 2.2 \\times 2.2\\\\ \\\\2.2^1 = 1 \\times 2.2\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><msup><mn>2.2<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mn>2.2<\/mn><mo>\u00d7<\/mo><mn>2.2<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><msup><mn>2.2<\/mn><mn>1<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mn>2.2<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">2.2^2 = 1 \\times 2.2 \\times 2.2\\\\ \\\\2.2^1 = 1 \\times 2.2<\/annotation><\/semantics><\/math><br><math data-latex=\"2.2^0 = 1 \\times \"><semantics><mrow><msup><mn>2.2<\/mn><mn>0<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2.2^0 = 1 \\times <\/annotation><\/semantics><\/math>Nothing (Because of 0, when there is nothing, 1 left alone always)<br><math display=\"block\"><semantics><mrow><msup><mn>2.2<\/mn><mn>0<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2.2^0 = 1<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(0) = a(1 + 2.2^b)<\/annotation><\/semantics><\/math><br>So the <strong>true y-intercept value<\/strong> is:<math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{a + a(2.2^b)}<\/annotation><\/semantics><\/math><br>Keep this exact form in mind \u2014 everything is compared to <strong>this value<\/strong>.<br><br><strong>\ud83e\udde0 STEP 3: Analyze Statement I CAREFULLY<\/strong><br>I. <math><semantics><mrow><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>x<\/mi><\/msup><mo>+<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">g(x) = a(2.2^x + k)<\/annotation><\/semantics><\/math><br>Because <math><semantics><mrow><mi>g<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">g<\/annotation><\/semantics><\/math> is equivalent to <math><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math>, we must have:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">k = 2.2^b<\/annotation><\/semantics><\/math><br>Now find the y-intercept of <math><semantics><mrow><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">g(x)<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">g(0) = a(1 + k)<\/annotation><\/semantics><\/math><br>Substitute <math><semantics><mrow><mi>k<\/mi><mo>=<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">k = 2.2^b<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">g(0) = a(1 + 2.2^b)<\/annotation><\/semantics><\/math><br>\u2714 This equals the correct y-intercept value.<br><br>\u26a0 <strong>BUT HERE IS THE KEY POINT (SAT LOGIC):<\/strong><br>~ The equation <strong>does NOT show<\/strong> <strong><math><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">a + a(2.2^b)<\/annotation><\/semantics><\/math><\/strong> explicitly<br>~ The y-intercept is <strong>hidden inside the symbol <\/strong><math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math><br>~ You cannot read the y-intercept <strong>directly<\/strong> from the equation<br>~ If the I takes common then it will match exactly the same to the equation but &#8220;The question especially emphasis on <strong>Display<\/strong> part,&#8221; so we cannot choose it.<br><br>Since:<br><math><semantics><mrow><mi>k<\/mi><mo>\u2260<\/mo><mi>a<\/mi><mo>+<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k \\neq a + a(2.2^b)<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> is just a placeholder, not the intercept itself<br>\u274c <strong>Statement I does NOT display the y-intercept as a constant or coefficient<\/strong><br><br><strong>\ud83e\udde0 STEP 4: Analyze Statement II (same strict logic)<\/strong><br>II. <math><semantics><mrow><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mn>2.2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>x<\/mi><\/msup><mo>+<\/mo><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h(x) = a(2.2)^x + m<\/annotation><\/semantics><\/math><br>Because <math><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math> is equivalent to <math><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math>, we must have:<math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m = a(2.2^b)<\/annotation><\/semantics><\/math><br>Now evaluate the y-intercept:<math display=\"block\"><semantics><mrow><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>m<\/mi><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><msup><mn>2.2<\/mn><mi>b<\/mi><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">h(0) = a(1) + m = a + a(2.2^b)<\/annotation><\/semantics><\/math><br>\u2714 This equals the correct y-intercept value.<br>\u26a0 <strong>But again \u2014 SAT precision matters:<\/strong><br>~ The y-intercept is <strong>not shown directly<\/strong><br>~ The constant <math><semantics><mrow><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math> equals <strong>only part<\/strong> of the intercept<br>~ The full intercept is <math><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a + m<\/annotation><\/semantics><\/math>, not just <math><semantics><mrow><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math><br>So:<br>~ The equation does <strong>not explicitly display<\/strong> the y-intercept<br>~ You must still compute it<br>\u274c <strong>Statement II also fails the requirement<\/strong><br><br><strong>\ud83e\udde0 STEP 5: Why \u201cI and II\u201d is WRONG (this is where I earlier erred<\/strong><br>The earlier mistake was assuming:<br>\u201cIf substituting x = 0 gives the intercept, then it is displayed.\u201d<br>That assumption is <strong>incorrect for this question<\/strong>.<br>The phrase <strong>\u201cdisplays the y-coordinate as a constant or coefficient\u201d<\/strong> means:<br>~ You should be able to <strong>see the y-intercept immediately<\/strong><br>~ Without substitution<br>~ Without hidden parameters<br>~ Without algebra<br>~ Neither equation does that.<br><strong>\u2705 CORRECT ANSWER: Option D<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mtext>Neither&nbsp;I&nbsp;nor&nbsp;II<\/mtext><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\text{Neither I nor II}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A seems possible but the word &#8220;Display&#8221; made it incorrect. The question is easy but it is a difficult category question, that means hidden keys. You must practice questions like these, so it won&#8217;t hinder your exam score.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is clearly incorrect, whether we talk about visual or equation, it never matches it<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is also incorrect because II is clearly incorrect.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>17th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question: <\/strong>Square A has side lengths that are 166 times the side lengths of square B. The area of square A is <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> times the area of square B. What is the value of <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math>?<br>A) 2<br>B) 166<br>C) 332<br>D) 27556<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\u2705 <strong>Understand the QUESTION \u2014 Squares &amp; Area Scaling<\/strong><br><strong>Given<\/strong><br>~ Side of square A = <strong>166 \u00d7<\/strong> side of square B<br>~ Area of square A = <math><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> \u00d7 area of square B<br>~ Find <math><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math><br><br><em><strong>1\ufe0f\u20e3 Key Rule (VERY IMPORTANT)<\/strong><br><\/em>\ud83d\udccc Area Scaling Rule<br>If side length scales by factor <math><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>,<math display=\"block\"><semantics><mrow><mtext>Area&nbsp;scales&nbsp;by&nbsp;<\/mtext><msup><mi>n<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Area scales by } n^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Calculation<\/strong><br>Side ratio:<math display=\"block\"><semantics><mrow><mn>166<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">166<\/annotation><\/semantics><\/math><br>Area ratio:<math display=\"block\"><semantics><mrow><msup><mn>166<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>27556<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">166^2 = 27556<\/annotation><\/semantics><\/math><br>\u2705 <strong>Correct Answer: Option D<\/strong><\/p>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>\u274c Why Other Options Are Wrong<\/strong><br>\u274c <strong>2<\/strong> \u2192 ignored scale completely<br>\u274c <strong>166<\/strong> \u2192 linear scaling mistake<br>\u274c <strong>332<\/strong> \u2192 doubled instead of squared<br>SAT <strong>loves squaring traps<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Trick (Legit Use)<\/strong><br>Type: 166^2<br>Instant verification.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>18th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% are parents of students, 45% are teachers from the current high school, 25% are school and district administrators, and the remaining 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?<br>A) 6<br>B) 8<br>C) 10<br>D) 18<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\u2705 <strong>Understand the Question<\/strong><br>Committee members:<br>~ 15% parents<br>~ 45% teachers<br>~ 25% administrators<br>~ Remaining = <strong>6 students<\/strong><br>How many <strong>more teachers than administrators<\/strong> were invited?<br><br>1\ufe0f\u20e3 What the Question Is Testing<br>This is <strong>percent \u2192 total \u2192 comparison<\/strong>.<br>Key challenge:<br>The <strong>total number is NOT given directly<\/strong> \u2014 you must find it.<br><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Solution<\/strong><br>Step 1: Add known percentages<math display=\"block\"><semantics><mrow><mn>15<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>+<\/mo><mn>45<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>+<\/mo><mn>25<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>85<\/mn><mi mathvariant=\"normal\">%<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">15\\% + 45\\% + 25\\% = 85\\%<\/annotation><\/semantics><\/math><br>So remaining:<math display=\"block\"><semantics><mrow><mn>100<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u2212<\/mo><mn>85<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>15<\/mn><mi mathvariant=\"normal\">%<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">100\\% &#8211; 85\\% = 15\\%<\/annotation><\/semantics><\/math><br>This 15% corresponds to <strong>6 students<\/strong>. Because they are <strong>remaining<\/strong>.<br><br>Step 2: Find the total number invited<math display=\"block\"><semantics><mrow><mn>15<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u2192<\/mo><mn>6<\/mn><mo>\u21d2<\/mo><mn>1<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u2192<\/mo><mfrac><mn>6<\/mn><mn>15<\/mn><\/mfrac><mo>=<\/mo><mn>0.4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">15\\% \\rightarrow 6 \\Rightarrow 1\\% \\rightarrow \\frac{6}{15} = 0.4<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>100<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u2192<\/mo><mn>0.4<\/mn><mo>\u00d7<\/mo><mn>100<\/mn><mo>=<\/mo><mn>40<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">100\\% \\rightarrow 0.4 \\times 100 = 40<\/annotation><\/semantics><\/math><br>Total invited = <strong>40 people<\/strong><br><br>Step 3: Find teachers and administrators<br>Teachers:<math display=\"block\"><semantics><mrow><mn>45<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u00d7<\/mo><mn>40<\/mn><mo>=<\/mo><mn>18<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">45\\% \\times 40 = 18<\/annotation><\/semantics><\/math><br>Administrators:<math display=\"block\"><semantics><mrow><mn>25<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u00d7<\/mo><mn>40<\/mn><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">25\\% \\times 40 = 10<\/annotation><\/semantics><\/math><br>Step 4: Compare<math display=\"block\"><semantics><mrow><mn>18<\/mn><mo>\u2212<\/mo><mn>10<\/mn><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">18 &#8211; 10 = 8<\/annotation><\/semantics><\/math><br><strong>\u2705 Correct Answer: Option B<br>8 more teachers<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\">\u274c Incorrect Options Explained<br>\u274c <strong>6<\/strong> \u2192 forgot to convert percent to total<br>\u274c <strong>10<\/strong> \u2192 confused administrators count with difference<br>\u274c <strong>18<\/strong> \u2192 gave number of teachers, not the difference<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Trick (REAL USE)<br><\/strong>1. Calculate all percentage:<br>~ Type: 100 &#8211; (15 + 45 +25)<br>~ Output: 15<br>That is out 15% which is equal to 6 students.<br>2. Type: 6 * 100\/15 <br>\u2192 gives <strong>40<\/strong><br>3. Then use teacher and administers percentage:<br>45 * 40\/100<br>0.25 * 40\/100<br>4. The output will be: 18  and 10<br>5. Find how many teachers are more than administers: 18 &#8211; 10<br>6. Output: 8<br>Fast and SAT-legal.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>19th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"149\" height=\"134\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2025\/12\/image.png\" alt=\"The solution of concept explanations of Algebra, Linear equations in two variables\" class=\"wp-image-8311\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The table gives the coordinates of two points on a line in the <math data-latex=\"xy\"><semantics><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">xy<\/annotation><\/semantics><\/math>-plane. The <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-intercept of the line is <math data-latex=\"(k - 5, b)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(k &#8211; 5, b)<\/annotation><\/semantics><\/math>, where <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are constants. What is the value of <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><br><strong>\u2705 Correct Answer: <math><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = 33<\/annotation><\/semantics><\/math><\/strong><br><br><strong>\ud83e\uddee Correct Solution \u2014 Step by Step<\/strong><br><strong><em>Step 1: Identify the two given points<br><\/em><\/strong>From the table:<br>Point 1 (<em>x<\/em><sub>1<\/sub>, <em>y<\/em><sub>1<\/sub>): <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(k,\\,13)<\/annotation><\/semantics><\/math><br>Point 2 (<em>x<\/em><sub>2<\/sub>, <em>y<\/em><sub>2<\/sub>): <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>+<\/mo><mn>7<\/mn><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mo>\u2212<\/mo><mn>15<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(k+7,\\,-15)<\/annotation><\/semantics><\/math><br><br><strong><em>Step 2: Find the slope of the line<\/em><\/strong><br>Slope formula:<br><math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mfrac><mrow><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><\/mrow><mrow><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">m = \\frac{y_2 &#8211; y_1}{x_2 &#8211; x_1}<\/annotation><\/semantics><\/math><br>Substitute:<math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>13<\/mn><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>+<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">m = \\frac{-15 &#8211; 13}{(k+7) &#8211; k}<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>28<\/mn><\/mrow><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">m = \\frac{-28}{7}<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = -4<\/annotation><\/semantics><\/math><br>\u2714 Slope is <strong>\u22124<\/strong><br><br><strong><em>Step 3: Use slope to find the y-intercept<\/em><\/strong><br>Use <strong>point-slope reasoning<\/strong> with point <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo separator=\"true\">,<\/mo><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(k, 13)<\/annotation><\/semantics><\/math>. Instead of using Point 1, you can also use Point 2.<br>Slope-intercept form:<math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = mx + b<\/annotation><\/semantics><\/math><br>Substitute values:<math display=\"block\"><semantics><mrow><mn>13<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">13 = -4(k) + b<\/annotation><\/semantics><\/math><br>Solve for <math><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>13<\/mn><mo>+<\/mo><mn>4<\/mn><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b = 13 + 4k<\/annotation><\/semantics><\/math><br><br><strong><em>Step 4: Use the given y-intercept coordinate<\/em><\/strong><br>The y-intercept is given as:<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mtext>\u2009<\/mtext><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(k &#8211; 5,\\, b)<\/annotation><\/semantics><\/math><br>But <strong><em>y<\/em>-intercept always has <em>x<\/em> = 0<\/strong>.<br>So:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k &#8211; 5 = 0<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k = 5<\/annotation><\/semantics><\/math><br><strong><em>Step 5: Substitute <\/em><\/strong><math><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k = 5<\/annotation><\/semantics><\/math> <strong><em>into<\/em><\/strong> <math><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>13<\/mn><mo>+<\/mo><mn>4<\/mn><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b = 13 + 4k<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>13<\/mn><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">b = 13 + 4(5)<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>13<\/mn><mo>+<\/mo><mn>20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = 13 + 20<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = 33<\/annotation><\/semantics><\/math><br><strong>\u2714 Correct value of <\/strong><math><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> is <strong>33<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Trap 1: b = 13 \u274c<\/strong><br><strong>How students get this:<\/strong><br>They think: \u201cThe y-intercept must be one of the given y-values.\u201d<br>But neither given point is at x = 0.<br>\u274c Incorrect logic<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Clarification<br><\/strong>First, what is a <em>y-intercept<\/em>? (This matters for Step 4)<br>A <strong>y-intercept<\/strong> is: <strong>The point where the graph crosses the y-axis<\/strong><br>This is not optional, not a formula trick \u2014 it is a <strong>definition<\/strong>.<br><br><strong>CRITICAL FACT<br><\/strong>On the <strong>y-axis<\/strong>, the x-coordinate is <strong>always 0<\/strong>.<br>Why?<br>Because:<br>~ The y-axis is the vertical line where x does not move left or right.<br>~ Any point on the y-axis has <strong>x = 0<\/strong>.<br>Examples:<br>~ (0, 5) \u2192 on the y-axis<br>~ (0, \u22123) \u2192 on the y-axis<br>~ (0, b) \u2192 y-intercept in general<br>If <strong>x \u2260 0<\/strong>, the point is <strong>not<\/strong> on the y-axis.<br><br><strong>\ud83e\uddee DESMOS CALCULATOR \u2014 CLEAN &amp; FAST METHOD<\/strong><br><strong>Step-by-Step in Desmos<\/strong><br>1. Open <strong>Desmos<\/strong><br>2. Define points:<br> ~ A = (5, 13)<br> ~ B = (12, -15)<br>3. Type:  y = -4x + b<br>4. Adjust <strong>b slider<\/strong> until the line passes through both points<br>5. Desmos shows: <em>b<\/em> = 33<br>\u2714 Visual confirmation<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>20th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> If <math data-latex=\"\\frac{\\sqrt{x^5}}{\\sqrt[3]{x^4}} = x^{\\frac{a}{b}}\"><semantics><mrow><mfrac><msqrt><msup><mi>x<\/mi><mn>5<\/mn><\/msup><\/msqrt><mroot><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mn>3<\/mn><\/mroot><\/mfrac><mo>=<\/mo><msup><mi>x<\/mi><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\sqrt{x^5}}{\\sqrt[3]{x^4}} = x^{\\frac{a}{b}}<\/annotation><\/semantics><\/math> for all positive values of <math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>, what is the value of <math data-latex=\"\\frac{a}{b}\"><semantics><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{a}{b}<\/annotation><\/semantics><\/math>?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-info\">1\ufe0f\u20e3 <strong>First: What is this \u201c3\u201d? Is it multiplied, divided, or something else?<\/strong><br>That <strong>small 3<\/strong> is <strong>NOT multiplication<\/strong>.<br>It is the <strong>index of the radical<\/strong>, meaning:<br>\ud83d\udd39 Rule<math display=\"block\"><semantics><mrow><mroot><msup><mi>x<\/mi><mi>m<\/mi><\/msup><mi>n<\/mi><\/mroot><mo>=<\/mo><msup><mi>x<\/mi><mfrac><mi>m<\/mi><mi>n<\/mi><\/mfrac><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[n]{x^m} = x^{\\frac{m}{n}}<\/annotation><\/semantics><\/math><br>So:<br>\u2022 Square root (a normal root like this <math data-latex=\"\\sqrt{x^5}\"><semantics><msqrt><msup><mi>x<\/mi><mn>5<\/mn><\/msup><\/msqrt><annotation encoding=\"application\/x-tex\">\\sqrt{x^5}<\/annotation><\/semantics><\/math>) \u2192 index or <math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> is <strong>2<\/strong><br>\u2022 Cube root (a cube root like this <math data-latex=\"\\sqrt[3]{x^4}\"><semantics><mroot><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mn>3<\/mn><\/mroot><annotation encoding=\"application\/x-tex\">\\sqrt[3]{x^4}<\/annotation><\/semantics><\/math>) \u2192 index or <math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> is <strong>3<\/strong><br><br>2\ufe0f\u20e3 <strong>Why Do We Use <\/strong><math data-latex=\"\\frac{m}{2}\"><semantics><mfrac><mi>m<\/mi><mn>2<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{m}{2}<\/annotation><\/semantics><\/math><strong> or <\/strong><math data-latex=\"\\frac{m}{n}\"><semantics><mfrac><mi>m<\/mi><mi>n<\/mi><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{m}{n}<\/annotation><\/semantics><\/math> <strong>?<\/strong><br>Because <strong>radicals and exponents are equivalent forms<\/strong>.<br>Fundamental Identity:<math display=\"block\"><semantics><mrow><mroot><msup><mi>x<\/mi><mi>m<\/mi><\/msup><mi>n<\/mi><\/mroot><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mi>m<\/mi><\/msup><msup><mo stretchy=\"false\">)<\/mo><mrow><mn>1<\/mn><mi mathvariant=\"normal\">\/<\/mi><mi>n<\/mi><\/mrow><\/msup><mo>=<\/mo><msup><mi>x<\/mi><mrow><mi>m<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>n<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[n]{x^m} = (x^m)^{1\/n} = x^{m\/n}<\/annotation><\/semantics><\/math><br>That\u2019s why:<br>\u2022 Square root \u2192 power <math><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{2}<\/annotation><\/semantics><\/math><br>\u2022 Cube root \u2192 power <math><semantics><mrow><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{3}<\/annotation><\/semantics><\/math><br><\/p>\n\n\n\n<p class=\"is-style-success\">3\ufe0f\u20e3 <strong>Step-by-Step Solution<\/strong><br><strong>Rewrite Each Part Using Exponents<\/strong><br>Numerator<math display=\"block\"><semantics><mrow><msqrt><msup><mi>x<\/mi><mn>5<\/mn><\/msup><\/msqrt><mo>=<\/mo><msup><mi>x<\/mi><mrow><mn>5<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{x^5} = x^{5\/2}<\/annotation><\/semantics><\/math><br>Explanation:<br>\u2022 Square root = power <math><semantics><mrow><mn>1<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1\/2<\/annotation><\/semantics><\/math><br>\u2022 So exponent becomes <math><semantics><mrow><mn>5<\/mn><mo>\u00d7<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>5<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">5 \\times \\frac{1}{2} = \\frac{5}{2}<\/annotation><\/semantics><\/math><br><br>Denominator<math display=\"block\"><semantics><mrow><mroot><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mn>3<\/mn><\/mroot><mo>=<\/mo><msup><mi>x<\/mi><mrow><mn>4<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>3<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{x^4} = x^{4\/3}<\/annotation><\/semantics><\/math><br>Explanation:<br>\u2022 Cube root = power <math><semantics><mrow><mn>1<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1\/3<\/annotation><\/semantics><\/math><br>\u2022 So exponent becomes <math><semantics><mrow><mn>4<\/mn><mo>\u00d7<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>4<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">4 \\times \\frac{1}{3} = \\frac{4}{3}<\/annotation><\/semantics><\/math><br><br>4\ufe0f\u20e3 <strong>Divide Powers with the Same Base<\/strong><br>We now have:<math display=\"block\"><semantics><mrow><mfrac><msup><mi>x<\/mi><mrow><mn>5<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><\/msup><msup><mi>x<\/mi><mrow><mn>4<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>3<\/mn><\/mrow><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{x^{5\/2}}{x^{4\/3}}<\/annotation><\/semantics><\/math><br>Rule:<br>What Divide is actually: We subtract values in divide, right?<br>So:<br><math display=\"block\"><semantics><mrow><mfrac><msup><mi>x<\/mi><mi>m<\/mi><\/msup><msup><mi>x<\/mi><mi>n<\/mi><\/msup><\/mfrac><mo>=<\/mo><msup><mi>x<\/mi><mrow><mi>m<\/mi><mo>\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{x^m}{x^n} = x^{m-n}<\/annotation><\/semantics><\/math><br><strong><em>Step-by-Step Subtraction<br><\/em><\/strong>Convert to a common denominator:<br><math data-latex=\"\\frac{x^\\frac{5}{2}}{x^\\frac{4}{3}}\\\\ \\\\ x^{\\frac{5}{2} - \\frac{4}{3}}\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mfrac><msup><mi>x<\/mi><mfrac><mn scriptlevel=\"2\">5<\/mn><mn scriptlevel=\"2\">2<\/mn><\/mfrac><\/msup><msup><mi>x<\/mi><mfrac><mn scriptlevel=\"2\">4<\/mn><mn scriptlevel=\"2\">3<\/mn><\/mfrac><\/msup><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><msup><mi>x<\/mi><mrow><mfrac><mn>5<\/mn><mn>2<\/mn><\/mfrac><mo>\u2212<\/mo><mfrac><mn>4<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\frac{x^\\frac{5}{2}}{x^\\frac{4}{3}}\\\\ \\\\ x^{\\frac{5}{2} &#8211; \\frac{4}{3}}<\/annotation><\/semantics><\/math><br><br>Take LCM of denominator 2 and 3: <math data-latex=\"\\frac{5}{2} - \\frac{4}{3}\"><semantics><mrow><mfrac><mn>5<\/mn><mn>2<\/mn><\/mfrac><mo>\u2212<\/mo><mfrac><mn>4<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{5}{2} &#8211; \\frac{4}{3}<\/annotation><\/semantics><\/math><br>It is 6.<br>Now divide New LCM Denominator by Old denominators:<br><math data-latex=\"6 \\div 2 = 3\\\\ \\\\6 \\div 3 = 2\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mn>6<\/mn><mo>\u00f7<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>6<\/mn><mo>\u00f7<\/mo><mn>3<\/mn><mo>=<\/mo><mn>2<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">6 \\div 2 = 3\\\\ \\\\6 \\div 3 = 2<\/annotation><\/semantics><\/math><br><br>Multiply the value to its numerator:<br><math data-latex=\"\\frac{5 \\times 3 - 4 \\times 2}{6}\"><semantics><mfrac><mrow><mn>5<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><\/mrow><mn>6<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{5 \\times 3 &#8211; 4 \\times 2}{6}<\/annotation><\/semantics><\/math><br><br>Subtract: <br><math data-latex=\"\\frac{15 - 8}{6} \\\\ \\frac{7}{6}\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>8<\/mn><\/mrow><mn>6<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mn>7<\/mn><mn>6<\/mn><\/mfrac><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\frac{15 &#8211; 8}{6} \\\\ \\frac{7}{6}<\/annotation><\/semantics><\/math><br><br>5\ufe0f\u20e3 <strong>Final Exponent<\/strong><math display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mfrac><mn>7<\/mn><mn>6<\/mn><\/mfrac><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^{\\frac{7}{6}}<\/annotation><\/semantics><\/math><br><math data-latex=\"x^\\frac{a}{b} = x^\\frac{7}{6}\"><semantics><mrow><msup><mi>x<\/mi><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><\/msup><mo>=<\/mo><msup><mi>x<\/mi><mfrac><mn>7<\/mn><mn>6<\/mn><\/mfrac><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^\\frac{a}{b} = x^\\frac{7}{6}<\/annotation><\/semantics><\/math><br>So: <math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mfrac><mi>a<\/mi><mi>b<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>7<\/mn><mn>6<\/mn><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\frac{a}{b} = \\frac{7}{6}}<\/annotation><\/semantics><\/math><br>\u2705 <strong>FINAL ANSWER: <\/strong><math data-latex=\"\\frac{7}{6}\"><semantics><mfrac><mn>7<\/mn><mn>6<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{7}{6}<\/annotation><\/semantics><\/math> or <math data-latex=\"1.166\"><semantics><mn>1.166<\/mn><annotation encoding=\"application\/x-tex\">1.166<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-warning\">\u26a0\ufe0f <strong>COMMON STUDENT MISTAKES (VERY IMPORTANT)<\/strong><br>\u274c Mistake 1: Treating the 3 as multiplication<br>Students think:<math display=\"block\"><semantics><mrow><msqrt><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>4<\/mn><\/msup><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{3x^4}<\/annotation><\/semantics><\/math><br>\u274c Completely wrong<\/p>\n\n\n\n<p class=\"is-style-warning\">\u274c Mistake 2: Writing <math><semantics><mrow><mroot><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mn>3<\/mn><\/mroot><mo>=<\/mo><msup><mi>x<\/mi><mrow><mn>4<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt[3]{x^4} = x^{4\/2}<\/annotation><\/semantics><\/math><br>Wrong \u2014 <strong>index controls denominator<\/strong><\/p>\n\n\n\n<p class=\"is-style-warning\">\u274c Mistake 3: Dividing exponents instead of subtracting<br>They do:<math display=\"block\"><semantics><mrow><mfrac><mrow><mn>5<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>2<\/mn><\/mrow><mrow><mn>4<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>3<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{5\/2}{4\/3}<\/annotation><\/semantics><\/math>\u274c Illegal operation<\/p>\n\n\n\n<p class=\"is-style-warning\">\u274c Mistake 4: Forgetting common denominator<br>Leads to wrong fraction like <math><semantics><mrow><mfrac><mn>1<\/mn><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{6}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\">\ud83e\uddee <strong>REAL DESMOS CHECK<\/strong><br>You must know basic concepts. All the Hard questions requires your Mathematical knowledge. You should use Desmos only as a calculator.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>21th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A sample of 40 fourth-grade students was selected at random from a certain school. The 40 students completed a survey about the morning announcements, and 32 thought the announcements were helpful. Which of the following is the largest population to which the results of the survey can be applied?<br>A) The 40 students who were surveyed<br>B) All fourth-grade students at the school<br>C) All students at the school<br>D) All fourth-grade students in the county in which the school is located<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">1\ufe0f\u20e3 What This Question Is Testing<br>~ sample of 40 fourth-grade students<br>~ sample from one school<br>~ largest population<br><br>This is <strong>statistical inference<\/strong>.<br>Key SAT rule:<br>You can generalize results <strong>only to the population from which the sample was randomly selected<\/strong>.<br><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Identify the Population<br><\/strong>~ Sample: <strong>40 fourth-graders<\/strong><br>~ Location: <strong>one school<\/strong><br>~ Random selection: <strong>within that school only<\/strong><br><br><strong>3\ufe0f\u20e3 Correct Answer Explained<\/strong><br>\u2705 <strong>B) All fourth-grade students at the school<\/strong><br>Why?<br>~ Same grade<br>~ Same school<br>~ Same population source<\/p>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>4\ufe0f\u20e3 Why Other Options Are Wrong<\/strong><br>\u274c <strong>A) Only the 40 students<\/strong><br>\u2192 Too narrow, 40 students of what \/ where (SAT wants the largest valid population)<br>\u274c <strong>C) All students at the school<\/strong><br>\u2192 Includes other grades \u2192 not sampled<br>\u274c <strong>D) All fourth-graders in the county<\/strong><br>\u2192 Different schools \u2192 not randomly sampled<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Note<\/strong><br>Desmos is <strong>not used<\/strong> here because:<br>~ No calculation required<br>~ This is a <strong>logic + sampling<\/strong> question<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>22th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math> is defined by <math data-latex=\"f(x) = -a^x + b\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msup><mi>a<\/mi><mi>x<\/mi><\/msup><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = -a^x + b<\/annotation><\/semantics><\/math>, where <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are constants. In the <math data-latex=\"xy\"><semantics><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">xy<\/annotation><\/semantics><\/math>-plane, the graph of <math data-latex=\"y = f(x) - 12\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = f(x) &#8211; 12<\/annotation><\/semantics><\/math> has a <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-intercept at <math data-latex=\"(0, -\\frac{75}{7})\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(0, -\\frac{75}{7})<\/annotation><\/semantics><\/math>. The product of <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> is <math data-latex=\"\\frac{320}{7}\"><semantics><mfrac><mn>320<\/mn><mn>7<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{320}{7}<\/annotation><\/semantics><\/math>. What is the value of <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\u2705 <strong>Understand the QUESTION<\/strong><br><strong>Question<\/strong><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><msup><mi>a<\/mi><mi>x<\/mi><\/msup><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = -a^x + b<\/annotation><\/semantics><\/math><br>The graph of:<math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = f(x) &#8211; 12<\/annotation><\/semantics><\/math><br>has a y-intercept at:<math display=\"block\"><semantics><mrow><mo fence=\"true\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\left(0, -\\frac{75}{7}\\right)<\/annotation><\/semantics><\/math><br>Given:<math display=\"block\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mfrac><mn>320<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">ab = \\frac{320}{7}<\/annotation><\/semantics><\/math><br>Find <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>.<br><br><strong>1\ufe0f\u20e3 Key Concept<br><\/strong>The <strong>y-intercept<\/strong> occurs when:<math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Solution<br><\/strong>Rather then first solving for <math data-latex=\"f(x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> then put the value of <math data-latex=\"f(x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> into the graph of <strong>y<\/strong>. It would be better that we directly put <math data-latex=\"f(x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> into the equation like below:<br><strong>Compute the y-Intercept Algebraically<\/strong><br>Start with:<math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup><mi>a<\/mi><mi>x<\/mi><\/msup><mo>+<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = -a^x + b &#8211; 12<\/annotation><\/semantics><\/math><br>Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup><mi>a<\/mi><mn>0<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = -a^0 + b &#8211; 12<\/annotation><\/semantics><\/math><br>Since:<br><math data-latex=\"a^2 = 1 \\times a \\times a\\\\ \\\\a^1 = 1 \\times a\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mi>a<\/mi><mo>\u00d7<\/mo><mi>a<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><msup><mi>a<\/mi><mn>1<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mi>a<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">a^2 = 1 \\times a \\times a\\\\ \\\\a^1 = 1 \\times a<\/annotation><\/semantics><\/math><br><br><math data-latex=\"a^0 = 1 \\times \"><semantics><mrow><msup><mi>a<\/mi><mn>0<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">a^0 = 1 \\times <\/annotation><\/semantics><\/math>Nothing, so it leaves with 1 only. <math data-latex=\"a^0 = 1\"><semantics><mrow><msup><mi>a<\/mi><mn>0<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a^0 = 1<\/annotation><\/semantics><\/math><br><br>Put the value of 1 in the equation: <math data-latex=\"y = -1 + b - 12\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = -1 + b &#8211; 12<\/annotation><\/semantics><\/math><br><math data-latex=\"y = b -1 - 12\\\\ \\\\y = b -13\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>y<\/mi><mo>=<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>12<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>y<\/mi><mo>=<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mn>13<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">y = b -1 &#8211; 12\\\\ \\\\y = b -13<\/annotation><\/semantics><\/math><br><br>3\ufe0f\u20e3 Use the Given y-Intercept: <math data-latex=\"(x, y) = (0, -\\frac{75}{7})\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x, y) = (0, -\\frac{75}{7})<\/annotation><\/semantics><\/math><br><br>Put the value of y-intercept now<br><math data-latex=\"-\\frac{75}{7} = b - 13\"><semantics><mrow><mo>\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo>=<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{75}{7} = b &#8211; 13<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>b<\/mi><mo>\u2212<\/mo><mn>13<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b &#8211; 13 = -\\frac{75}{7}<\/annotation><\/semantics><\/math><br>Add 13 to both sides or just move <strong>-13<\/strong> to right-hand side:<br>Let&#8217;s see both ways for educational purpose:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Add 13 to both sides<\/th><th class=\"has-text-align-center\" data-align=\"center\">Move -13 to Right-hand side<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b - 13 = -\\frac{75}{7}\"><semantics><mrow><mi>b<\/mi><mo>\u2212<\/mo><mn>13<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b &#8211; 13 = -\\frac{75}{7}<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b - 13 = -\\frac{75}{7}\"><semantics><mrow><mi>b<\/mi><mo>\u2212<\/mo><mn>13<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b &#8211; 13 = -\\frac{75}{7}<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b - 13 + 13 = -\\frac{75}{7} + 13\"><semantics><mrow><mi>b<\/mi><mo>\u2212<\/mo><mn>13<\/mn><mo>+<\/mo><mn>13<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo>+<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b &#8211; 13 + 13 = -\\frac{75}{7} + 13<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = -\\frac{75}{7} + \\frac{13}{1}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>13<\/mn><mn>1<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = -\\frac{75}{7} + \\frac{13}{1}<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = -\\frac{75}{7} + \\frac{13}{1}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>75<\/mn><mn>7<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>13<\/mn><mn>1<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = -\\frac{75}{7} + \\frac{13}{1}<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = \\frac{-75 \\times 1 + 13 \\times 7}{7}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>75<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>+<\/mo><mn>13<\/mn><mo>\u00d7<\/mo><mn>7<\/mn><\/mrow><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = \\frac{-75 \\times 1 + 13 \\times 7}{7}<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = \\frac{-75 \\times 1 + 13 \\times 7}{7}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>75<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>+<\/mo><mn>13<\/mn><mo>\u00d7<\/mo><mn>7<\/mn><\/mrow><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = \\frac{-75 \\times 1 + 13 \\times 7}{7}<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = \\frac{-75 + 91}{7}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>75<\/mn><mo>+<\/mo><mn>91<\/mn><\/mrow><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = \\frac{-75 + 91}{7}<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = \\frac{-75 + 91}{7}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>75<\/mn><mo>+<\/mo><mn>91<\/mn><\/mrow><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = \\frac{-75 + 91}{7}<\/annotation><\/semantics><\/math><\/td><\/tr><\/tbody><tfoot><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"b = \\frac{16}{7}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mfrac><mn>16<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">b = \\frac{16}{7}<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\" b = \\frac{16}{7}\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mfrac><mn>16<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"> b = \\frac{16}{7}<\/annotation><\/semantics><\/math><\/td><\/tr><\/tfoot><\/table><\/figure>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\">4\ufe0f\u20e3 Use the Product Condition<br>We have <strong>b<\/strong> now and we need to find <strong>a<\/strong><br><math display=\"block\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mfrac><mn>320<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">ab = \\frac{320}{7}<\/annotation><\/semantics><\/math><br>Substitute <math><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>\u22c5<\/mo><mfrac><mn>16<\/mn><mn>7<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>320<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a \\cdot \\frac{16}{7} = \\frac{320}{7}<\/annotation><\/semantics><\/math><br>Multiply both sides by 7:<br><math data-latex=\"\\frac{16a}{7} \\times 7 = \\frac{320}{7} \\times 7\\\\ \\\\16a = 320\\\\ \\\\ a = \\frac{320}{16} \\\\ \\\\a = 20\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mrow><mn>16<\/mn><mi>a<\/mi><\/mrow><mn>7<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>7<\/mn><mo>=<\/mo><mfrac><mn>320<\/mn><mn>7<\/mn><\/mfrac><mo>\u00d7<\/mo><mn>7<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>16<\/mn><mi>a<\/mi><mo>=<\/mo><mn>320<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mn>320<\/mn><mn>16<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mo>=<\/mo><mn>20<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\frac{16a}{7} \\times 7 = \\frac{320}{7} \\times 7\\\\ \\\\16a = 320\\\\ \\\\ a = \\frac{320}{16} \\\\ \\\\a = 20<\/annotation><\/semantics><\/math><br><br>\u2705 <strong>Correct Answer: 20<\/strong><\/p>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\">\u274c Typical Traps<br>\u2022 Forgetting to subtract 12<br>\u2022 Using <math><semantics><mrow><msup><mi>a<\/mi><mn>0<\/mn><\/msup><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a^0 = 0<\/annotation><\/semantics><\/math> instead of 1<br>\u2022 Mixing up intercept with asymptote<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\ud83e\uddee <strong>Desmos Confirmation<\/strong><br>Use Desmos only for calculator, above you are given all steps, just understand them and crack it up.<\/p>\n<\/div><\/details><\/div>\n<\/div>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>Did you try all the features and get comfortable using them? You should work on using the calculator and seeing references and directions. So be prepared for everything before taking the final SAT exam. The explanation of answers makes easier to learn and progress. You must try to work on your speed and spend less time on the beginning and more on the later questions. This is the SAT 2024 Practice Test of Math Module 2nd.<\/p>\n\n\n\n<p>There are more tests available:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-math-test-4-module-2nd-preparation\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT 2025 Test (Math Module 1st)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-math-test-3-module-1st-study-guide\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 3rd (Math Module 1st)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-reading-and-writing-test-6-module-2nd\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 6th (Reading and Writing Module 2nd)<\/a><\/li>\n<\/ul>\n\n\n\n<p>The best way to become a master in Math is to find the correct answer and understand why other options are incorrect. I wish you luck in your bright career.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>SAT Math 2024 Module 2nd (How to Get 1500+ Hack, Free Test: The SAT real practice test of 2024 exam &#8211; Math Module 2nd &#8211; all four options explained deeply with Math tricks &#038; Desmos hack. First you take the test then learn<\/p>\n","protected":false},"author":1,"featured_media":8629,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"googlesitekit_rrm_CAowmvTFDA:productID":"","_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","footnotes":""},"categories":[13,16],"tags":[24,27,29],"class_list":["post-8302","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-2nd-module","category-sat-2024","tag-sat-2024","tag-sat-math","tag-sat-module-2nd"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/8302","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/comments?post=8302"}],"version-history":[{"count":2,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/8302\/revisions"}],"predecessor-version":[{"id":8899,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/8302\/revisions\/8899"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/media\/8629"}],"wp:attachment":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/media?parent=8302"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/categories?post=8302"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/tags?post=8302"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}