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<h4>Activity (15 minutes)</h4>
<p>Students continue to compare quadratic and exponential functions in this activity. This time, they decide how to
compare the functions.</p>
<p>Monitor for different strategies students may use to compare the functions. Identify students who:</p>
<ul>
<li> Create one or more tables of values and compare the values of \(p(x)\) and \(q(x)\) at increasingly large values
of \(x\). </li>
<li> Graph the functions defined by \(p(x)=6x^2\) and \(q(x)=3^x\) and compare the graphs. </li>
<li> Create one or more tables of values, calculate the growth factors at equal intervals of input, and then compare
the growth factors. </li>
</ul>
<p>If students choose graphing or spreadsheet tools strategically to perform comparisons, they are demonstrating an
understanding of the benefits of different strategies.</p>
<h4>Launch</h4>
<p>Ask students to observe the equations representing the two functions and determine which function is exponential and
which is quadratic. Invite students to share how they know. Make sure students recognize that \(p\) is quadratic and
\(q\) is exponential before they begin the activity.</p>
<p>Provide access to graphing technology and spreadsheet tools. This may be a good opportunity for students to
experiment with the graphing window. If the horizontal dimension is very small (for example, \(0<x<5\)) or the
vertical dimension is very large (for example,\(0<y<3,000\)), the two graphs will be hard to distinguish. As
needed, remind students to think about adjusting the graphing window to make the graphs more informative.</p>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 2 Collect and Display: Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p> As students share their analysis with the class, listen for and
collect the language students use to identify and describe how the output of the exponential function eventually
outgrows that of the quadratic function. Write the students' words and phrases on a visual display and update it
throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will
help students read and use mathematical language during their partner and whole-group discussions.</p>
<p class="os-raise-text-italicize"> Design
Principle(s): Maximize meta-awareness</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Here are two functions: \(p(x)=6x^2\) and \(q(x)=3^x\).</p>
<p>Investigate the output of \(p\) and \(q\) for different values of \(x\). For large enough values of \(x\), one
function will have a greater value than the other.</p>
<p>Which function will have a greater value as \(x\) increases?</p>
<p>Support your answer with tables, graphs, or other representations.</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> \(q\) is larger when \(x=0\) and again when \(x=5\). Once \(x\) is larger than \(5\), \(q\)
is growing much more quickly, as shown in the table.</p>
<table class="os-raise-doubleheadertable">
<thead>
<tr>
<th scope="col"> </th>
<th scope="col">\(p(x)\)</th>
<th scope="col">\(q(x)\)</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">\(0\)</th>
<td>\(0\)</td>
<td>\(1\)</td>
</tr>
<tr>
<th scope="row">\(1\)</th>
<td>\(6\)</td>
<td>\(3\)</td>
</tr>
<tr>
<th scope="row">\(2\)</th>
<td>\(24\)</td>
<td>\(9\)</td>
</tr>
<tr>
<th scope="row">\(3\)</th>
<td>\(54\)</td>
<td>\(27\)</td>
</tr>
<tr>
<th scope="row">\(4\)</th>
<td>\(96\)</td>
<td>\(81\)</td>
</tr>
<tr>
<th scope="row">\(5\)</th>
<td>\(150\)</td>
<td>\(243\)</td>
</tr>
<tr>
<th scope="row">\(6\)</th>
<td>\(216\)</td>
<td>\(729\)</td>
</tr>
<tr>
<th scope="row">\(7\)</th>
<td>\(294\)</td>
<td>\(2,187\)</td>
</tr>
</tbody>
</table>
<br>
<p>Alternatively, here is a graph showing the plotted values from the table and another showing \(y=p(x)\) and
\(y=q(x)\).</p>
<p><img alt="A graph showing the plotted values from the table." height="199" src="https://k12.openstax.org/contents/raise/resources/b97213610fd3c72fe39f2131e5e5e063b9d38db1" width="284"></p>
<br>
<p><img alt="Graphs of q and p on same graph." height="204" src="https://k12.openstax.org/contents/raise/resources/23262d28b2176518cfc60ae1e8b2aa4c23a77992" width="290"></p>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<ol class="os-raise-noindent">
<li>Janna says that some exponential functions grow more slowly than the quadratic function as \(x\) increases. Do you
agree with Janna? Be prepared to show your reasoning.</li>
</ol>
<p><strong>Answer:</strong> Yes. This is true for the functions \(g(x)=(1 \frac {1}{2})x\) and \(f(x)=x^2\) as \(x\)
increases for \(x\)-values less than \(10\).</p>
<ol class="os-raise-noindent" start="2">
<li>Could you have an exponential function \(g(x)=b^x\) so that \(g(x)<f(x)\) for all values of \(x\)?</li>
</ol>
<p><strong>Answer:</strong> No, \(g(0)>f(0)\) for all values of \(b\) (except \(0\)), because \(b^0=1\) and
\(0^2=0\). (If \(b=0\), there are problems if \(x\leq 0\), so this is not usually considered an exponential function.)
</p>
<h4>Video: Comparing Exponential and Quadratic Functions</h4>
<p>Watch the following video to learn more about functions.</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/a3e459a0bf5aade786458e33de808ab06fbd6993">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/b565aa6188289f4173de077195d4ebf7fd904c9b" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/a3e459a0bf5aade786458e33de808ab06fbd6993
</video></div>
</div>
<br>
<br>
<h4>Activity Synthesis</h4>
<p>Select students to present their strategies in the sequence listed in the Activity Narrative. If no students chose to
graph the functions, consider displaying the graphs for all to see. By presenting strategies that involve the
calculation of growth factors last, students will deepen their understanding of why exponential values will eventually
overtake the quadratic values.</p>
<p>Again, one way to make sense of why exponential functions eventually grow faster than quadratic functions is by
thinking of the growth factors. The output of an exponential function always increases by the same factor when the
input increases by \(1\) (for example, \(3\) for the exponential function \(q\) studied here). The output of a
quadratic function, in contrast, increases by smaller and smaller amounts when the input increases by \(1\). So even
though a quadratic function may take larger values than an exponential function for many inputs, the values of the
exponential function will eventually overtake the quadratic function.</p>
<h3>7.4.3: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts
explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Which of the following has the greatest value as \(x\) increases?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(y=4 \cdot x^2\)</td>
<td>Incorrect. Let’s try again a different way: This quadratic function will increase as \(x\) increases but
will not have the greatest value when compared to an exponential growth function. The answer is \(y=4^x\).</td>
</tr>
<tr>
<td>\(y=4^x\)</td>
<td>That’s correct! Check yourself: This is an exponential function with a growth factor of \(4\), which
will have the greatest value as \(x\) increases.</td>
</tr>
<tr>
<td>\(y=4x\)</td>
<td>Incorrect. Let’s try again a different way: This function is linear and has a constant difference of 4,
but that will not increase as much as an exponential function. The answer is \(y=4^x\).</td>
</tr>
<tr>
<td>\(y=(\frac {1}{4})^x\)</td>
<td>Incorrect. Let’s try again a different way: This has a growth factor of \(\frac {1}{4}\), which will
decay, not increase. The answer is \(y=4^x\).</td>
</tr>
</tbody>
</table>
<br>
<h3>7.4.3: Additional Resources</h3>
<em><strong>
<p>The following content is available to students who would like more support based on their experience with the
self check. Students will not automatically have access to this content, so you may wish to share it with those
who could benefit from it.</p>
</strong></em>
<h4>Exponential versus Quadratic Functions</h4>
<p>For each of the two functions, do the following:</p>
<ol class="os-raise-noindent" type="a">
<li> Complete the table of values. </li>
<li> Sketch a graph. </li>
<li> Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know. </li>
</ol>
<ol class="os-raise-noindent">
<li> \(f(x)=3\cdot 2^x\)</li>
</ol>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">\(x\)</th>
<td>\(-1\)</td>
<td>\(0\)</td>
<td>\(1\)</td>
<td>\(2\)</td>
<td>\(3\)</td>
<td>\(5\)</td>
</tr>
<tr>
<th scope="row">\(f(x)\)</th>
<td>\(\frac{3}{2}\) or equivalent </td>
<td>\(3\)</td>
<td>\(6\)</td>
<td>\(12\)</td>
<td>\(24\)</td>
<td>\(96\)</td>
</tr>
</tbody>
</table>
<br>
<p><img height="298" src="https://k12.openstax.org/contents/raise/resources/88922da47465d7b4e2be357a2e6d1c8b87bb093b" width="300"></p>
<p>This function is exponential. It has a growth factor of \(2\).</p>
<ol class="os-raise-noindent" start="2">
<li>\(g(x)=3\cdot x^2\)</li>
</ol>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">\(x\)</th>
<td>\(-1\)</td>
<td>\(0\)</td>
<td>\(1\)</td>
<td>\(2\)</td>
<td>\(3\)</td>
<td>\(5\)</td>
</tr>
<tr>
<th scope="row">\(g(x)\)</th>
<td>\(3\) </td>
<td>\(0\)</td>
<td>\(3\)</td>
<td>\(12\)</td>
<td>\(27\)</td>
<td>\(75\)</td>
</tr>
</tbody>
</table>
<br>
<p><img height="304" src="https://k12.openstax.org/contents/raise/resources/c9c483e847a71d346d069f57164ded7e77f81809" width="300"></p>
<p>This function is not linear because it does not have a constant rate of change, and it is not exponential because it
does not have a constant growth factor. The function is quadratic because it can be represented by \(g(x)=3x^2\),
which has \(x\) to the second power, and that is the greatest power.</p>
<p>Note: Between \(f(x)\) and \(g(x)\), \(f(x)\) will have the greater value as \(x\) increases.</p>
<h4>Try It: Exponential versus Quadratic Functions</h4>
<p>Which of the following has the greater value as \(x\) increases, \(f(x)\) or \(g(x)\)?</p>
<p>\(f(x)=3x^2\)</p>
<p>\(g(x)=3^x\)</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to determine which function has the greater value as \(x\) increases:</p>
<p>Make a table of values for each function, and then graph the function:</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">\(x\)</th>
<td>\(-2\)</td>
<td>\(-1\)</td>
<td>\(0\)</td>
<td>\(1\)</td>
<td>\(2\)</td>
<td>\(3\)</td>
<td>\(4\)</td>
</tr>
<tr>
<th scope="row">\(f(x)=3x^2\)</th>
<td>\(12\)</td>
<td>\(3\)</td>
<td>\(0\)</td>
<td>\(3\)</td>
<td>\(12\)</td>
<td>\(27\)</td>
<td>\(48\)</td>
</tr>
</tbody>
</table>
<br>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="304" role="presentation" src="https://k12.openstax.org/contents/raise/resources/f4631cb9d02c271990fc27002a5536212ecd6a16" width="300"></p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">\(x\)</th>
<td>\(-2\)</td>
<td>\(-1\)</td>
<td>\(0\)</td>
<td>\(1\)</td>
<td>\(2\)</td>
<td>\(3\)</td>
<td>\(4\)</td>
</tr>
<tr>
<th scope="row">\(g(x)=3^x\)</th>
<td>\(\frac {1}{9}\)</td>
<td>\(\frac {1}{3}\)</td>
<td>\(1\)</td>
<td>\(3\)</td>
<td>\(9\)</td>
<td>\(27\)</td>
<td>\(81\)</td>
</tr>
</tbody>
</table>
<br>
<p><img alt class="img-fluid atto_image_button_text-bottom" role="presentation" src="https://k12.openstax.org/contents/raise/resources/6ead768f506e99b9144290b0b041f973f9159381" width="260"></p>
<p>Looking at both the table of values and the graphs, \(g(x)\) will have a greater value than \(f(x)\) as \(x\)
increases. This is because \(g(x)\) is exponential and has a constant growth factor.</p>