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d058b6c2-dc36-4f9c-a04e-077466053f34.html
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<h4>Activity</h4>
<p>Consider two functions defined by \(f(x)=x(x+4)\) and \(g(x)=x(x-4)\).</p>
<ol class="os-raise-noindent">
<li>Consider the function defined by \(f(x)=x(x+4)\).</li>
</ol>
<p>Then, answer questions a–g, and create a table of values in which to enter the given x values and their outputs using the function.</p>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="60e03d00-5cf3-4cbe-9995-126ce55591d8" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1a-->
<div class="os-raise-ib-pset-problem" data-content-id="fbf538bb-f2c6-4352-9d03-9245f6291f4a" data-problem-comparator="integer" data-problem-type="input" data-solution="0">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> When the value of \(x=-4\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 0</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 0.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1b-->
<div class="os-raise-ib-pset-problem" data-content-id="7e3b60e8-2698-4351-b519-0dbf05268d6c" data-problem-comparator="integer" data-problem-type="input" data-solution="-3">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>When the value of \(x=-3\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! -3</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is -3.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1c-->
<div class="os-raise-ib-pset-problem" data-content-id="36eff251-576a-4a08-a292-23ebe6b0caed" data-problem-comparator="integer" data-problem-type="input" data-solution="0">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> When the value of \(x=0\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 0</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 0.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1d-->
<div class="os-raise-ib-pset-problem" data-content-id="c7ec1749-f6a9-420d-b76f-5177b09b99d9" data-problem-comparator="integer" data-problem-type="input" data-solution="5">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li>When the value of \(x=1\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 5</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 5.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1e-->
<div class="os-raise-ib-pset-problem" data-content-id="4ead2d8e-cb7a-4723-884e-3f093a5cfd92" data-problem-comparator="integer" data-problem-type="input" data-solution="12">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li> When the value of \(x=2\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 12</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 12.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1f-->
<div class="os-raise-ib-pset-problem" data-content-id="3e2149b6-ffc9-4aa2-b353-5138c9cac85c" data-problem-comparator="integer" data-problem-type="input" data-solution="21">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="6" type="a">
<li> When the value of \(x=3\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 21</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 21.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1g-->
<div class="os-raise-ib-pset-problem" data-content-id="d7ee71b0-6b6c-47ba-9a7a-331832b8b0b4" data-problem-comparator="integer" data-problem-type="input" data-solution="45">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="7" type="a">
<li> When the value of \(x=5\), what is the value of \(f(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 45</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 45.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<ol class="os-raise-noindent" start="2">
<li>Now, answer questions a–d to analyze the function defined by \(f(x)=x(x+4)\).</li>
</ol>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ea5dd96d-2c3a-41cf-9101-4a6ee6b2f10f" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> What are the \(x\)-intercepts of the graph? Be prepared to show your reasoning.</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>The \(x\)-intercepts for function \(f\) are \((-4,0)\) and \((0,0)\). These points have an \(x\)-coordinate that makes the output value 0.</p>
</div>
</div>
<!--Interaction End -->
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="5b09d861-2c42-4fc9-adac-3b0c9323c015" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1a-->
<div class="os-raise-ib-pset-problem" data-content-id="cd4fd094-81f0-44ce-bdc2-7c8ffd4a284c" data-problem-comparator="integer" data-problem-type="input" data-solution="-2">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the value of the \(x\)-coordinate in the vertex of the graph?</li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! -2</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is -2.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1b-->
<div class="os-raise-ib-pset-problem" data-content-id="efc5a9b1-58bb-48f7-afa5-e4c45538c19e" data-problem-comparator="integer" data-problem-type="input" data-solution="-4">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What is the value of the \(y\)-coordinate in the vertex of the graph?</li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! -4</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is -4.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ff30d984-f026-461c-88d8-f0ff819533c7" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> What is the vertex of the graph of function \(f\)? Be prepared to show your reasoning.</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> The vertex appears to be at \((-2,-4)\). From the tables, the function values seem to change from decreasing to increasing at those points.</p>
</div>
</div>
<!--Interaction End -->
<ol class="os-raise-noindent" start="3">
<li>Consider the function defined by \(g(x)=x(x-4)\).<br>
<br>
Then, answer questions a–g, and create a table of values in which to enter the given x values and their outputs using the function.
</li>
</ol>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="b62c56c4-8444-4b97-b412-cf0455dc0e67" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1a-->
<div class="os-raise-ib-pset-problem" data-content-id="e2b10c58-e3be-4468-b659-41230961de8d" data-problem-comparator="integer" data-problem-type="input" data-solution="32">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> When the value of \(x=-4\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 32</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 32.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1b-->
<div class="os-raise-ib-pset-problem" data-content-id="0ded0a1a-af60-483d-8d79-07df7ca15ea9" data-problem-comparator="integer" data-problem-type="input" data-solution="21">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> When the value of \(x=-3\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 21</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 21.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1c-->
<div class="os-raise-ib-pset-problem" data-content-id="22c89b68-2f83-45e6-9397-cb6cccd55df5" data-problem-comparator="integer" data-problem-type="input" data-solution="0">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> When the value of \(x=0\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 0</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 0.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1d-->
<div class="os-raise-ib-pset-problem" data-content-id="fc365c44-9ee1-4f0b-b7f9-4e1165326aad" data-problem-comparator="integer" data-problem-type="input" data-solution="-3">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> When the value of \(x=1\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! -3</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is -3.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1e-->
<div class="os-raise-ib-pset-problem" data-content-id="dcbf8f49-f691-4ee2-b84d-5e96b1cfd5d4" data-problem-comparator="integer" data-problem-type="input" data-solution="-4">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li> When the value of \(x=2\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! -4</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is -4.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1f-->
<div class="os-raise-ib-pset-problem" data-content-id="570ad872-751f-4c87-b55b-f71032c93d75" data-problem-comparator="integer" data-problem-type="input" data-solution="0">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="6" type="a">
<li> When the value of \(x=4\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 0</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 0.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q1g-->
<div class="os-raise-ib-pset-problem" data-content-id="5b0b1aff-8af2-4e7b-9fdc-0c41da2f0930" data-problem-comparator="integer" data-problem-type="input" data-solution="5">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="7" type="a">
<li> When the value of \(x=5\), what is the value of \(g(x)\)? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 5</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 5.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<ol class="os-raise-noindent" start="4">
<li>Now, answer questions a–d to analyze the function defined by \(g(x)=x(x-4)\).</li>
</ol>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="dc14e538-d2a4-4ae5-b464-ee8c4f59e8d6" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> What are the \(x\)-intercepts of the graph? Be prepared to show your reasoning.</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>The \(x\)-intercepts for function \(g\) are \((0,0)\) and \((4,0)\). These points have an \(x\)-coordinate that makes the output value 0.</p>
</div>
</div>
<!--Interaction End -->
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="7b816718-4c6d-4f06-93e9-d7ca103afdf3" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1a-->
<div class="os-raise-ib-pset-problem" data-content-id="3192dc06-76cb-40d2-9e9b-1ead9c311094" data-problem-comparator="integer" data-problem-type="input" data-solution="2">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the value of the \(x\)-coordinate in the vertex of the graph? </li>
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</ol>
<p>Enter your answer here:<strong><br></strong></p>
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<div class="os-raise-ib-pset-correct-response">
<p>Correct! 2</p>
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<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 2.</p>
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<div class="os-raise-ib-pset-problem" data-content-id="94ee4a87-d64f-4f3a-ad32-31ba6177aa8b" data-problem-comparator="integer" data-problem-type="input" data-solution="-4">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What is the value of the \(y\)-coordinate in the vertex of the graph? </li>
</ol>
</ol>
<p>Enter your answer here:<strong><br></strong></p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! -4</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is -4.</p>
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<div class="os-raise-ib-pset-encourage-response">
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="f4c81000-da59-4f8e-beb4-0b8bf0740b69" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> What is the vertex of the graph of function \(g\)? Be prepared to show your reasoning. </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> The vertex appears to be at \((2,-4)\). From the tables, the function values seem to change from decreasing to increasing at those points.</p>
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</div>
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<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<ol class="os-raise-noindent" start="5">
<li>Use the graphing tool or technology outside the course.</li>
</ol>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Plot the points from the tables on the same coordinate plane. (Consider using different colors or markings for each set of points so you can tell them apart.) </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p><img alt="Coordinate grid with two graphs shown by plotted points." height="267" src="https://k12.openstax.org/contents/raise/resources/1817dcc93434ecbd070c0706316edaae60d639c3" width="262"></p>
</div>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="7cd352e7-006f-4385-927c-407ac73884f3" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Make a couple of observations about how the two graphs compare. </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<ul>
<li> The \(x\)-intercept of both graphs is at \((0,0)\). </li>
<li> The graphs seem to be a reflection of each other across the \(y\)-axis. </li>
<li> The vertices both have a \(y\)-coordinate of -4 and also mirror one another across the \(y\)-axis. </li>
<li> Each function mirrors itself across a vertical line through its vertex. </li>
</ul>
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