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8878640a-9401-4baa-83ba-40b86ec02c94.html
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<h4>Activity</h4>
<ol class="os-raise-noindent">
<li>Here is one way to rewrite \(3x^2+12x+9\) in <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">vertex form (of a quadratic expression)</span>.</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" type="a">
<li>Study the steps and write a brief explanation of what is happening at each step.</li>
</ol>
<p>Original expression<br>
\(3x^2+12x+9\)</p>
<p><strong>Step 1</strong> - \(3(x^2+4x+3)\)</p>
<p><strong>Step 2</strong> - \(3(x^2+4x+4+3−4)\)</p>
<p><strong>Step 3</strong> - \(3(x^2+4x+4−1)\)</p>
<p><strong>Step 4</strong> - \(3((x+2)^2−1)\)</p>
<p><strong>Step 5</strong> - \(3(x+2)^2−3\)</p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then 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 answers:</p>
<p>Original expression<br>
\(3x^2+12x+9\)</p>
<p><strong>Step 1</strong> - Rewrite the expression with 3 as a factor.<br>
\(3(x^2+4x+3)\)</p>
<p><strong>Step 2</strong> - Add 4 to complete the square.<br>
Subtract 4 to keep the value of the expression the same.<br>
\(3(x^2+4x+4+3−4)\)</p>
<p><strong>Step 3</strong> - Combine like terms so we can see the perfect square, \(x^2+4x+4\).<br>
\(3(x^2+4x+4−1)\)</p>
<p><strong>Step 4</strong> - Rewrite \(x^2+4x+4\) as a squared expression.<br>
\(3((x+2)^2−1)\)</p>
<p><strong>Step 5</strong> - Apply the distributive property to put the expression in vertex form.<br>
\(3(x+2)^2−3\)</p>
</div>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="b71640c3-23f2-4186-98e0-dd901768db6b" 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="94449039-e496-4222-9357-adb316d75f6e" data-problem-comparator="integer" data-problem-type="input" data-solution="-2">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>What is the \(x\)-coordinate of the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">vertex (of a graph)</span> that represents this expression?</li>
</ol>
<p>Enter the \(x\)-coordinate of the vertex. </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The \(x\)-coordinate of the vertex is -2.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is -2. The \(x\)-coordinate of the vertex is -2.</p>
</div>
</div>
<!--END QUESTION.-->
<br>
<!--Q1b-->
<div class="os-raise-ib-pset-problem" data-content-id="f18e0bfe-9bca-49e0-9491-9ba2737676f3" data-problem-comparator="integer" data-problem-type="input" data-solution="-3">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>What is the \(y\)-coordinate of the vertex of the graph that represents this expression?</li>
</ol>
<p>Enter the \(y\)-coordinate of the vertex. </p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The \(y\)-coordinate of the vertex is -3.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is -3. The \(y\)-coordinate of the vertex is -3.</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>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="4" type="a">
<li>Does the graph open upward or downward? Explain how you know.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then 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="Reveal2">
<p>Compare your answer:</p>
<p> The graph opens upward. For example, in an earlier unit, we saw that when the coefficient of the squared term
\((x−h)^2\) is positive, the graph opens upward.</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal3" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>Rewrite \(-2x^2−4x+6\) in vertex form.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then 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="Reveal3">
<p>Compare your answer:</p>
<p>\(-2(x+1)^2+8\)</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal4" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="3">
<li>Rewrite \(4x^2+24x+20\) in vertex form.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then 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="Reveal4">
<p>Compare your answer:</p>
<p>\(4(x+3)^2−16\)</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal5" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="4">
<li>Rewrite \(-x^2+20x\) in vertex form.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then 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="Reveal5">
<p>Compare your answer:</p>
<p>\(-(x−10)^2+100\)</p>
</div>
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="78c77769-6f93-4c0b-9054-002d575f8e0c" data-retry-limit="0" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem" data-content-id="ab222ab2-5d97-49df-8acc-336fae8e8aa4" data-problem-comparator="float" data-problem-type="input" data-solution="2">
<div class="os-raise-ib-pset-problem-content">
<h4>Extending Your Thinking</h4>
<ol class="os-raise-noindent">
<li>Fill in the missing parts of the function \(f(x)=2(x−3)(x−9)\) in vertex form \(f(x)=
\underline{\;\mathbf a\boldsymbol.\;} (x \underline{\;\mathbf b\boldsymbol.\;} \; \underline{\;\mathbf
c\boldsymbol.\;} )^2 \underline{\;\mathbf d\boldsymbol.\;}\; \underline{\;\mathbf e\boldsymbol.\;} \) without
completing the square. (Hint: Think about finding the zeros of the function.)</li>
<ol class="os-raise-noindent" type="a">
<li>Enter the value of a.</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 2</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is 2.</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="277518af-2d92-470c-ac83-5496e448abd3" data-problem-type="multiplechoice" data-solution="Subtraction(-)" data-solution-options='["Subtraction(-)", "Addition(+)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>Enter the sign of b.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Minus (-)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is minus (-)</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="8cd977bb-ada8-403c-8e9c-4081c87392f6" data-problem-comparator="float" data-problem-type="input" data-solution="6">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Enter the value of c.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 6</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is 6.</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="910680bf-69a5-4f48-96ca-d0497eb75ee5" data-problem-type="multiplechoice" data-solution="Subtraction(-)" data-solution-options='["Subtraction(-)", "Addition(+)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="4" type="a">
<li>Enter the sign of d.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Minus (-)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is minus (-)</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="fef2672c-3097-4539-a128-fbe4c56b1a3a" data-problem-comparator="float" data-problem-type="input" data-solution="18">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="5" type="a">
<li>Enter the value of e.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 18</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is 18.</p>
</div>
</div>
<!--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>
<!--End interaction-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="0910840a-be68-4b9e-9a4b-bf6ee8dc9825" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6" type="a">
<li>Be prepared to show your reasoning for your answers.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your explanation. </p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>The zeros of \(f\) are 3 and 9. Because the graph of a quadratic function has a vertical line of symmetry across
the vertex, the vertex of the graph must have \(x\)-coordinate 6. The \(y\)-coordinate of the vertex is -18
because \(f(6)=-18\). This means \(f(x)\) can be expressed as \(a(x−6)^2−18\). If we expand
\(2(x−3)(x−9)\), we can see that the coefficient of \(x^2\) is 2. This means that \(a\) in the vertex
form has to be 2 as well.</p>
</div>
</div>
<!--End interaction-->
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="6a97e420-812f-47e7-a0d6-2a9f7ee437b6" data-retry-limit="0" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem" data-content-id="02de0598-1fa0-4520-ba6e-37d0ac19e22d" data-problem-comparator="float" data-problem-type="input" data-solution="2">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2">
<li>Fill in the missing parts of the function \(g(x)=2(x−3)(x−9)+21\) in vertex form \(g(x)=
\underline{\;\mathbf a\boldsymbol.\;} (x \underline{\;\mathbf b\boldsymbol.\;} \;\underline{\;\mathbf
c\boldsymbol.\;} )^2 \underline{\;\mathbf d\boldsymbol.\;} \;\underline{\;\mathbf e\boldsymbol.\;}\) without
completing the square.</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Enter the value of a.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 2</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is 2.</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="fba80e14-fdbf-4f0d-8703-583700f83fd9" data-problem-type="multiplechoice" data-solution="Subtraction(-)" data-solution-options='["Subtraction(-)", "Addition(+)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>Enter the sign of b.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Minus (-)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is minus (-)</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="d3a81070-2192-4244-b09c-b289636dca8c" data-problem-comparator="float" data-problem-type="input" data-solution="6">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Enter the value of c.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 6</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is 6.</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="01da4daa-5e72-4557-9a2e-2df9526aa50b" data-problem-type="multiplechoice" data-solution="Addition(+)" data-solution-options='["Subtraction(-)", "Addition(+)"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="4" type="a">
<li>Enter the sign of d.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Plus (+)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is plus (+)</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="e362abd6-cf29-48ab-b377-849f9629b625" data-problem-comparator="float" data-problem-type="input" data-solution="3">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="5" type="a">
<li>Enter the value of e.</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 3</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is 3.</p>
</div>
</div>
<!--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>
<!--End interaction-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="64f884ec-169a-4e05-9e38-e5f1f76a8040" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6" type="a">
<li>Be prepared to show your reasoning for your answers.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your explanation. </p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Although 3 and 9 are not zeros of \(g\), when \(x\) is 3 and when \(x\) is 9, \(g(x)\) is 21, or
\(g(3)=g(9)=21\). Because the graphs of quadratic functions are symmetric, the \(x\)-coordinate of the vertex is
still 6. The \(y\)-coordinate of the vertex is 3 because \(g(6)=3\). The value of \(a\) is 2 for the same reason
as in function \(f\).</p>
</div>
</div>
<!--End interaction-->
<h4>Video: Rewriting Expressions in Vertex Form</h4>
<p>Watch the following video to learn more about rewriting expressions in vertex form.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/f4900a114d3f80ff7308afc6c3831e2d1db15424">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/05b54ce32b43e94121a9ff134d2c9640fcb197f1" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/f4900a114d3f80ff7308afc6c3831e2d1db15424
</video></div>
</div>