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<h4>Activity</h4>
<br>
<div class="os-raise-familysupport">
<p>Access the <a href="https://k12.openstax.org/contents/raise/resources/ad776e2f4b03a08bf69ddddc59a1efd31551b1a0" target="_blank">Desmos guide PDF</a> for tips on solving problems with the Desmos graphing calculator.</p>
</div>
<br>
<ol class="os-raise-noindent">
<li> Each of the following functions, \(f\), \(g\), \(h\), and \(j\), represents the amount of money in a bank account, in dollars, as a function of time \(x\), in years. They are each written in the form \(m(x)=a \cdot b^x\).
</li><br>
</ol>
<ul>
<li>\(f(x)=50 \cdot 2^x\)</li>
<li>\(g(x)=50 \cdot 3^x\)</li>
<li>\(h(x)=50 \cdot (\frac{3}{2})^x\)</li>
<li>\(j(x)=50 \cdot (0.5)^x\)</li>
</ul><br>
<ol class="os-raise-noindent" type="a">
<li>Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-bottom="-50" data-left="-1" data-right="5" data-schema-version="1.0" data-top="250"></div>
<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">
</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="Reveal1">
<p>Compare your answers:</p>
<p> <img alt height="328" role="presentation" src="https://k12.openstax.org/contents/raise/resources/65124031f2fe922dd130df9822a442d7f2e73334" width="300"></p>
</div>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="a8ad806a-835f-4a41-8158-2abd0c2ce10d" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>Explain how changing the value of \(b\)<em> </em>changes the graph.</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>When \(b\) is greater than 1, larger values of \(b\) mean that the function grows more quickly as \(x\) increases. A positive value of \(b\) that is less than 1 means the function is decreasing as \(x\) increases.</p>
</div>
</div>
<ol class="os-raise-noindent" start="2">
<li>Here are equations defining functions \(p\), \(q\), and \(r\). They are also written in the form \(m(x)=a \cdot b^x\).</li>
</ol><br>
<ul>
<li>\(p(x)=10 \cdot 4^x\)</li>
<li>\(q(x)=40 \cdot 4^x\)</li>
<li>\(r(x)=100 \cdot 4^x\)</li>
</ul><br>
<ol class="os-raise-noindent" type="a">
<li>Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-bottom="-50" data-left="-3" data-right="3" data-schema-version="1.0" data-top="250"></div>
<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">
</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 answers:</p>
<p> <img alt height="328" role="presentation" src="https://k12.openstax.org/contents/raise/resources/498752c95425675132f2709b3136d187da172b81" width="300"></p>
</div>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="750dce14-f3b3-49af-82d5-dac93bcb73e0" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>Explain how changing the value of \(a\) changes the graph.</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>It changes the \(y\)-intercept to \((0,a)\).</p>
</div>
</div>
<ol class="os-raise-noindent" start="3">
<li> Here are equations defining functions \(f\), \(g\), and \(h\) written in the form \(m(x)=a \cdot b^x+d\).</li>
</ol><br>
<ul>
<li>\(f(x)=4^x+2\)</li>
<li>\(g(x)=4^x\)</li>
<li>\(h(x)=4^x-2\)</li>
</ul><br>
<ol class="os-raise-noindent" type="a">
<li>Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-bottom="-7" data-left="-8" data-right="8" data-schema-version="1.0" data-top="8"></div>
<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">
</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 answers:</p>
<p> <img alt height="328" role="presentation" src="https://k12.openstax.org/contents/raise/resources/498752c95425675132f2709b3136d187da172b81" width="300"></p>
</div>
<ol class="os-raise-noindent" type="a">
<li>Explain how changing the value of \(a\) changes the graph.</li>
</ol>
<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">
</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 answers:</p>
<img alt="GRAPH OF THREE EXPONENTIAL GROWTH FUNCTIONS, LABELED F, G, AND H. FUNCTION G HAS A \(y\)-intercepts OF 1. FUNCTION F HAS A \(y\)-intercepts OF 3. FUNCTION H HAS A \(y\)-intercepts OF NEGATIVE 1." class="atto_image_button_text-bottom" height="309" src="https://k12.openstax.org/contents/raise/resources/1aa6001125c68e96e7c9379ee684dd09f0860b65" width="267">
</div>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="55815306-b733-4aab-8b98-81d2735e18f8" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2" type="a">
<li>Discuss with a partner how changing the value of \(d\) changes the asymptote of the graph.</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>Changing the value of \(d\) changes the horizontal asymptote from \(y=0\) to \(y=d\). So function \(f(x)\) has a horizontal asymptote of \(y= 2\), and \(h(x)\) has a horizontal asymptote of \(y= -2\).</p>
</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="c26ece26-8711-4b31-9dd8-d166106fcfa3" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<br>
<p>As before, consider bank accounts whose balances are given by the following functions:</p>
<p>\(f(x)=10 \cdot3^x\) \(g(x)=3^{x+2}\) \(h(x)=\frac{1}{2} \cdot 3^{x+3}\)</p>
<p>Which function would you choose? Does your choice depend on \(x\)?</p>
</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>We can rewrite each of the functions as multiples of \(3^x\) to more easily compare them:</p>
<p>\(f(x)=10 \cdot 3^x\)</p>
<p>\(g(x)=3^{x+2}=3^x \cdot 3^{2}=9 \cdot 3^x\)</p>
<p>\(h(x)=\frac{1}{2} \cdot 3^{x+3}=\frac{1}{2} \cdot 3x \cdot 3^3=\frac{27}{2} \cdot 3^x=(13.5) \cdot 3^x\)</p>
<p>Since \(9<10<13.5\), we see that \(h\) has the most money for any value of \(x\).</p>
</div>
</div>
<!--End interaction-->
<br>
<h4>Video: Exponential Equations and How They Affect the Graphs</h4>
<p>Watch the following video to learn more about exponential functions and how they affect the graphs:</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/c418a58864a702dfb9a80d1ba26aeeb7ce0b4724">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/ad14f93f793d28e90170608577c2194ca8ef93db" srclang="en_us">https://k12.openstax.org/contents/raise/resources/c418a58864a702dfb9a80d1ba26aeeb7ce0b4724
</video></div>
</div>
<br>
<br>