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<h4>Activity</h4>
<ol class="os-raise-noindent">
<li>The graph that represents \(p(x)=(x−8)^2+1\) has its <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">vertex (of a graph)</span> at \((8,1)\). Here is one way to show, without graphing, that \((8,1)\) corresponds to the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">minimum</span> value of \(p\).</li>
</ol>
<ul class="os-raise-noindent">
<li> When \(x=8\), the value of \((x−8)^2\) is 0 because \((8−8)^2=0^2=0\). </li>
<li> Squaring any number always results in a positive number, so when \(x\) is any value other than 8, \((x−8)\) will be a number other than 0, and when squared, \((x−8)^2\) will be positive. </li>
<li> Any positive number is greater than 0, so when \(x \neq 8\), the value of \((x−8)^2\) will be greater than when \(x=8\). In other words, \(p\) has the least value when \(x=8\). </li>
</ul>
<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">
<p>Use similar reasoning to explain why the point \((4,1)\) corresponds to the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">maximum</span> value of \(q\), defined by \(q(x)=-2(x−4)^2+1\).</p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Enter your reasoning here.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<ul class="os-raise-noindent">
<li> When \(x=4\), the value of \((x−4)^2\) is 0 because \(-2(4−4)^2=-2(0^2)=0\). </li>
<li> Squaring any number always results in a positive number, so when \(x\) is any value other than 4 (either greater or less), \((x−4)\) will be a number other than 0. When squared, \((x−4)^2\) will be positive, but then it gets multiplied to a negative number, so the product will be negative. </li>
<li> Any negative number is less than 0, so when \(x \neq 4\), the value of \(-2(x−4)^2\) will be less than when \(x=4\). In other words, \(q\) has the greatest value when \(x=4\). </li>
</ul>
</div>
<p>The greatest value of a function is the maximum value. The lowest value of a function is the minimum value. Since infinity cannot be reached or touched, it is not considered a maximum or minimum.</p>
<p>In quadratic functions, the maximum or minimum value occurs at the vertex of the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">parabola</span>.</p>
<ol class="os-raise-noindent" start="2">
<li>Here are some quadratic functions and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function.</li>
</ol>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">
Equation
</th>
<th scope="col">
Coordinates of the vertex
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
a.
</td>
<td>
\(f(x)=-(x−4)^2+6\)
</td>
<td>
\((4,6)\)
</td>
</tr>
<tr>
<td>
b.
</td>
<td>
\(g(x)=(x+7)^2−3\)
</td>
<td>
\((-7,-3)\)
</td>
</tr>
<tr>
<td>
c.
</td>
<td>
\(h(x)=4(x+5)^2+7\)
</td>
<td>
\((-5,7)\)
</td>
</tr>
<tr>
<td>
d.
</td>
<td>
\(k(x)=x^2−6x−3\)
</td>
<td>
\((3,-12)\)
</td>
</tr>
<tr>
<td>
e.
</td>
<td>
\(m(x)=-x^2+8x\)
</td>
<td>
\((4,16)\)
</td>
</tr>
</tbody>
</table>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="0c49a6c7-154e-45d4-9331-cf199e344e7e" 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="35f8f732-434d-4d07-9274-b87ec33765ee" data-problem-type="dropdown" data-solution="Maximum" data-solution-options='["Maximum", "Minimum"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" type="a">
<li>Determine if the vertex corresponds to the maximum or the minimum value of the function \(f(x)=-(x−4)^2+6\).</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is maximum. The leading coefficient is negative, so the parabola opens down.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p> Incorrect. The correct answer is maximum. The leading coefficient is negative, so the parabola opens down.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q2b-->
<div class="os-raise-ib-pset-problem" data-content-id="0485fc99-f31c-46b4-8d28-4d62ed99a810" data-problem-type="dropdown" data-solution="Minimum" data-solution-options='["Maximum", "Minimum"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>Determine if the vertex corresponds to the maximum or the minimum value of the function \(g(x)=(x+7)^2−3\).</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is minimum. The leading coefficient is positive, so the parabola opens up.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p> Incorrect. The correct answer is minimum. The leading coefficient is positive, so the parabola opens up.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q2c-->
<div class="os-raise-ib-pset-problem" data-content-id="da068d33-ced2-4d24-a7a9-0cab78d1f5bf" data-problem-type="dropdown" data-solution="Minimum" data-solution-options='["Maximum", "Minimum"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Determine if the vertex corresponds to the maximum or the minimum value of the function \(h(x)=4(x+5)^2+7\).</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is minimum. The leading coefficient is positive, so the parabola opens up.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p> Incorrect. The correct answer is minimum. The leading coefficient is positive, so the parabola opens up.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q2d-->
<div class="os-raise-ib-pset-problem" data-content-id="39283ad7-e025-4abc-bdfe-0f2a4f98d6af" data-problem-type="dropdown" data-solution="Minimum" data-solution-options='["Maximum", "Minimum"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="4" type="a">
<li>Determine if the vertex corresponds to the maximum or the minimum value of the function \(k(x)=x^2−6x−3\).</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is minimum. The leading coefficient is positive, so the parabola opens up.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p> Incorrect. The correct answer is minimum. The leading coefficient is positive, so the parabola opens up.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q2e-->
<div class="os-raise-ib-pset-problem" data-content-id="973a07e0-cf6d-4808-8a08-f933319bfa50" data-problem-type="dropdown" data-solution="Maximum" data-solution-options='["Maximum", "Minimum"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="5" type="a">
<li>Determine if the vertex corresponds to the maximum or the minimum value of the function \(m(x)=-x^2+8x\).</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is maximum. The leading coefficient is negative, so the parabola opens down.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p> Incorrect. The correct answer is maximum. The leading coefficient is negative, so the parabola opens down.</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>
<!-- "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-input" data-button-text="Solution" data-content-id="9530e886-c9dd-43a2-b0b6-54564c2d9c4d" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>Here is a portion of the graph of function \(q\), defined by \(q(x)=-x^2+14x−40\).</p>
<p><img height="182" src="https://k12.openstax.org/contents/raise/resources/41f2639468a13e21e3859a1761fc87e759a7bee0" width="254"></p>
<p>\(ABCD\) is a rectangle. Points \(A\) and \(B\) coincide with the \(x\)-intercepts of the graph, and segment \(CD\) just touches the vertex of the graph.</p>
<p>Find the area of \(ABCD\). Show your reasoning.</p>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the area of \(ABCD\) and your reasoning. </p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>54 square units.
</p>
<p>For example: </p>
<ul class="os-raise-noindent">
<li> The expression can be rewritten in factored form as \((-x+10)(x−4)\), so the \(x\)-intercepts are \((4,0)\) and \((10,0)\), which means the length of \(AB\) is 6 units. The \(x\)-coordinate of the vertex is halfway between 4 and 10, which is 7. Substituting 7 for \(x\) in \((- x + 10)(x − 4)\) gives \(3 \cdot 3\) or 9, so the height of the rectangle is 9. The area is \(6 \cdot 9\) or 54 square units. </li>
<li> The equation \(q(x)=-x^2+14x−40\) can be written in vertex form as \(q(x)=-1(x−7)^2+9\), so its vertex is at \((7,9)\) and the height of the rectangle is 9 units. Completing the square for \(-x^2+14x−40=0\) gives the solutions \(x=4\) and \(x=10\), which correspond to the \(x\)-intercepts at points \(A\) and \(B\). This means the width of the rectangle is \(10−4\) or 6 units. </li>
</ul>
</div>
</div>
<!--End interaction-->