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9c3a71f4-cd5a-4b03-baba-64d95e36300b.html
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<h4>Activity</h4>
<p>Here are two sets of equations for quadratic functions you saw earlier. In each set, the expressions that define the output are equivalent. In other words, all the equations in Set 1 define the same function. The same is true for Set 2: all the equations define the same function but they are just written in different forms.</p>
<p><strong>Set 1:</strong></p>
<p>\(f(x)=x^2+4x\)</p>
<p>\(g(x)=x(x+4)\)</p>
<p>\(h(x)=(x+2)^2−4\)</p>
<p><strong>Set 2:</strong></p>
<p>\(p(x)=-x^2+6x−5\)</p>
<p>\(q(x)=(5−x)(x−1)\)</p>
<p>\(r(x)=-1(x−3)^2+4\)</p>
<p>The expression that defines \(h\) is written in <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">vertex form</span>. We can show that it is equivalent to the expression defining \(f\) by expanding and simplifying the expression:</p>
<p>\(\begin{array}{rcl}(x+2)^2-4&=&(x+2)(x+2)-4\\&=&x^2+2x+2x+4-4\\&=&x^2+4x,\;\mathrm{which}\;\mathrm{is}\;f(x)\end{array}\) </p>
<p>If this simplified expression is factored, then we arrive at the expression that defines function \(g\): \(x^2+4x=x(x+4)\). The expression that defines \(g\) is written in <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">factored form</span>. The simplified expression that defines function \(f\) is written in <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">standard form</span>.</p>
<br>
<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">
<li>Show that the expressions defining \(r\) and \(p\) are equivalent.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer. Then select the <strong>s</strong><strong>olution</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>Expanding and simplifying function \(r\), \(-1(x−3)^2+4 gives -1(x^2−6x+9)+4\), which equals \(-x^2+6x−9+4\) or \(-x^2+6x−5\).</p>
</div>
<br>
<p>For questions 2–3, examine the graphs representing the quadratic functions.</p>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="90cc397a-32df-4908-897a-092366df8f59" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q2a-->
<div class="os-raise-ib-pset-problem" data-content-id="ffff5156-2bf7-4567-96b2-7aa7f7191d18" 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>Examine the graph of the function \(h\).</li>
</ol>
<p><img height="171" src="https://k12.openstax.org/contents/raise/resources/b19d8c50682c182199f7c90742727e13eb68319d" width="268"></p>
<ol class="os-raise-noindent" type="a">
<li>What is the \(x\)-coordinate of the vertex of the graph of \(h\)?</li>
</ol>
<p>Enter your answer here:</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.-->
<!--Q2b-->
<div class="os-raise-ib-pset-problem" data-content-id="467b3ca3-8550-4222-ba3f-a1c76865ea1b" data-problem-comparator="float" data-problem-type="input" data-solution="-4">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>What is the \(y\)-coordinate of the vertex of the graph of \(h\)?</li>
</ol>
<p>Enter your answer here:</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.-->
<!--Q3a-->
<div class="os-raise-ib-pset-problem" data-content-id="0773894f-eec9-475a-82ba-23df40c73c4f" 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="3">
<li>Examine the graph of the function \(r\).</li>
</ol>
<p><img height="171" src="https://k12.openstax.org/contents/raise/resources/1edb88678883c574d4b558c14b48a109300fe1e9" width="268"></p>
<ol class="os-raise-noindent" type="a">
<li> What is the \(x\)-coordinate of the vertex of the graph of \(r\)?</li>
</ol>
<p>Enter your answer here:</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.-->
<!--Q3b-->
<div class="os-raise-ib-pset-problem" data-content-id="abdf1679-d62b-4b66-bdb8-979b6c361039" data-problem-comparator="float" data-problem-type="input" data-solution="4">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the \(y\)-coordinate of the vertex of the graph of \(r\)?</li>
</ol>
<p>Enter your answer here:</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>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="786b9724-51ce-4a04-9083-60d32531949f" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>Why do you think expressions such as those defining \(h\) and \(r\) are said to be written in vertex form?</li>
</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 numbers in vertex form seem to give the coordinates of the vertex of the graph. When a positive number is added to the \(x\) in the parentheses, the \(x\)-coordinate of the vertex seems to be negative—which is the opposite of that number. When a number is subtracted from \(x\), the \(x\)-coordinate of the vertex seems to be positive—which is the opposite of what it looks like.</p>
</div>
</div>
<br>
<!--Interaction End -->
<div class="os-raise-student-reflection">
<p class="os-raise-student-reflection-title">Why Should I Care?</p>
<img src="https://k12.openstax.org/contents/raise/resources/668a0db00bbc24bbceaecf0a660d2ecd86729414" width="600">
<p>When you launch an object, it does not travel in a straight line. Instead, it follows a curved path called a parabola. This means you have to account for distance, height, and angle if you want to hit your target. You can use quadratic equations to find these measurements. </p>
<p>The next time you throw a ball, think about the parabolic path that it follows.</p>
</img>
</div>