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<h3>Warm Up (5 minutes)</h3>
<p>In this activity, students recall the meaning of maximum or minimum value of a function, which they learned in a previous unit. They also practice interpreting the language related to maximum and minimum values of functions.</p>
<h4>Launch</h4>
<p>Ask students to describe some situations in which people use the words "minimum" and "maximum." For example, we might say there is a minimum age for voting or for getting a driver's license, or that roads and highways have maximum speed limits.</p>
<p>Then, ask students what the words "minimum" and "maximum" mean more generally. We might think of a minimum as the least, the least possible, or the least allowable value, and a maximum as the greatest, the greatest possible, or the greatest allowable value.</p>
<h4>Student Activity </h4>
<p>Examine the graph that represents the function, \(f\), defined by \(f(x)=(x−4)^2+1\).</p>
<p><img height="291" src="https://k12.openstax.org/contents/raise/resources/4246a818d67db1e72a333d627b6aa12064c74adc" width="300"></p>
<p>Use this graph to answer questions 1 - 2.</p>
<ol class="os-raise-noindent">
<li>\(f(1)\) can be expressed in words as "the value of \(f\) when \(x\) is 1." Find or compute:</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li> the value of \(f\) when \(x\) is 1</li>
</ol>
<p><strong>Answer:</strong> 10</p>
<ol class="os-raise-noindent" start="2" type="a">
<li>\( f(3)\)</li>
</ol>
<p><strong>Answer:</strong> 2</p>
<ol class="os-raise-noindent" start="3" type="a">
<li> \(f(10)\)</li>
</ol>
<p><strong>Answer:</strong> 37</p>
<ol class="os-raise-noindent" start="2">
<li>Can you find an \(x\)-value that would make \(f(x)\):</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Less than 1?</li>
</ol>
<ul class="os-raise-noindent">
<li> Yes </li>
<li> No </li>
</ul>
<p><strong>Answer:</strong> No<br>
From the graph, it seems that 1 is the least value of \(f\), and that occurs when \(x=4\). All other \(x\)-values produce outputs that are greater than 1. </p>
<ol class="os-raise-noindent" start="2" type="a">
<li>Greater than 10,000?</li>
</ol>
<ul class="os-raise-noindent">
<li> Yes </li>
<li> No </li>
</ul>
<p><strong>Answer:</strong> Yes <br>
There are lots of \(x\)-values that produce an output greater than 10,000. For example, both -100 and 110 would make \(f(x)\) greater than 10,000.</p>
<p>Examine the graph that represents the function, \(g\), defined by \(g(x)=-(x−12)^2+7\).</p>
<p><img height="291" src="https://k12.openstax.org/contents/raise/resources/ce9c6929c1bd313d327c435153311f2fd64e328f" width="300"></p>
<p>Use this graph to answer questions 3 and 4.</p>
<ol class="os-raise-noindent" start="3">
<li>\(g(9)\) can be expressed in words as "the value of \(g\) when \(x\) is 9." Find or compute:</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>the value of \(g\) when \(x\) is 9</li>
</ol>
<p><strong>Answer:</strong> -2</p>
<ol class="os-raise-noindent" start="2" type="a">
<li>\(g(13)\)</li>
</ol>
<p><strong>Answer:</strong> 6</p>
<ol class="os-raise-noindent" start="3" type="a">
<li>\(g(2)\)</li>
</ol>
<p><strong>Answer:</strong> -93</p>
<ol class="os-raise-noindent" start="4">
<li>Can you find an \(x\)-value that would make \(g(x)\):</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Greater than 7?</li>
</ol>
<ul class="os-raise-noindent">
<li> Yes </li>
<li> No </li>
</ul>
<p><strong>Answer:</strong> No<br>
From the graph, it seems that 7 is the greatest value of \(g\), and that occurs when \(x=12\). All other \(x\)-values produce outputs that are less than 7. </p>
<ol class="os-raise-noindent" start="2" type="a">
<li>Less than -10,000?</li>
</ol>
<ul class="os-raise-noindent">
<li> Yes </li>
<li> No </li>
</ul>
<p><strong>Answer:</strong> Yes <br>
There are lots of \(x\)-values that produce an output less than -10,000. For example, both -200 and 200 would make \(g(x)\) greater than -10,000.</p>
<h4>Anticipated Misconceptions</h4>
<p>Some students may struggle to relate the \(y\)-coordinates of the points on a graph with the outputs of a function. Earlier in the course, students learned that the graph of a function \(f\) is the graph of the equation \(y=f(x)\). Consider having students label the coordinates of points on each graph and then complete the statements, such as, "The point \((3,2)\) on the graph means \(2=f(3)\)." Another approach would be to have students organize the points on the graphs into tables with headers \(x\) and \(f(x)\) and \(x\) and \(g(x)\).</p>
<h4>Activity Synthesis</h4>
<p>Discuss with students:</p>
<ul class="os-raise-noindent">
<li> "Why does \(f\) not have a maximum value?" (We can always use larger values of \(x\) in both the positive and negative directions to get greater and greater values of \(f\).) </li>
<li> "Why does \(g\) not have a minimum value?" (We can always find an input that makes the value of \(g\) less and less.) </li>
</ul>
<p>Emphasize that we can find an input that makes the value of \(f\) as great as we want and that makes \(g\) as small as we want.</p>
<p>Remind students that:</p>
<ul class="os-raise-noindent">
<li> A maximum value of a function is a value of a function that is greater than or equal to all the other values. It corresponds to the highest \(y\)-value on the graph of the function. </li>
<li> A minimum value of a function is a value of a function that is less than or equal to all the other values. It corresponds to the lowest point on the graph of the function. </li>
<li> For quadratic functions, there is only one maximum or minimum value. </li>
</ul>