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<h4>Activity (15 minutes)</h4>
<p>This activity complements the previous one. For a situation characterized by exponential growth, a larger base value
\(b\) means faster growth. For a situation characterized by exponential decay, however, a smaller value of \(b\)
(between 0 and 1) corresponds to faster decay. Understanding how the parameter \(b\) influences the graph of an
exponential function will prepare students to make sense of and model situations involving repeated percentage change
later in the unit.</p>
<h4>Launch</h4>
<p>Present the following scenario: “Suppose you are presented with four functions, \(m\), \(n\), \(p\), and
\(q\),<em> </em>that describe the amount of money, in dollars, in a bank account as a function of time \(x\), in
years. If the account is yours (and more money is better), which function would you choose? Why?”</p>
<p>Here are equations defining the functions.</p>
<p>\(m(x)=200 \cdot (\frac{1}{4})^x\)</p>
<p>\(n(x)=200 \cdot (\frac{1}{2})^x\)</p>
<p>\(p(x)=200 \cdot (\frac{3}{4})^x\)</p>
<p>\(q(x)=200 \cdot (\frac{7}{8})^x\)</p>
<p>Give students quiet time to think individually and ask students to share their responses. Ask a student who chose the
first option and one who chose the last option to share their reasoning. Tell students they will now consider the
options graphically before confirming their choice.</p>
<p>Arrange students in groups of 2. Provide access to graphing technology.</p>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title"> Support for English Language Learners</p>
</div>
<div class="os-raise-extrasupport-body">
<p>In groups of 2, students should take turns explaining their reasoning of how they matched the equations to the
graphs. Display the following sentence frames for all to see: “The equation _____ matches graph _____ because
. . .” and “I noticed _____ , so I matched . . . .” Encourage students to challenge each other
when they disagree. When monitoring the discussions, point out key words and phrases such as “the bases are
less than 1,” “slowly decaying,” or “more vertical curve.” This will help students
solidify their understanding of how the “b” affects the graph.</p>
</div>
</div>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Differentiate the degree of difficulty by beginning with more accessible values. When presenting the scenario,
begin by displaying only \(m(x)\) and \(p(x)\) for students to consider briefly before revealing the remaining
functions. In addition, some students may benefit from revisiting findings from the warm up, in which after spending
\(\frac13\) of the gift money, Jada had \(\frac23\) left. </p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
<li> Match each function with a graph. Be prepared to show your reasoning. </li>
</ol>
<p>\(m(x)=200 \cdot (\frac{1}{4})^x\)</p>
<p>\(n(x)=200 \cdot (\frac{1}{2})^x\)</p>
<p>\(p(x)=200 \cdot (\frac{3}{4})^x\)</p>
<p>\(q(x)=200 \cdot (\frac{7}{8})^x\)</p>
<p><img alt="Graph of four exponential function on x y grid. A. B. C. D."
src="https://k12.openstax.org/contents/raise/resources/d2271503c0dabaa6350b65982c8be4b0936b0260"></p>
<p><strong>Answer:</strong> For example: The functions are all exponential, and the bases are all less than 1, so the
graphs will be decreasing. The most rapidly decreasing will be the one with the smallest base, and the least rapidly
decreasing will be the one with the largest base. </p>
<p>Graph D: \(m(x)=200 \cdot (\frac{1}{4})^x\)</p>
<p>Graph C: \(n(x)=200 \cdot (\frac{1}{2})^x\)</p>
<p>Graph B: \(p(x)=200 \cdot (\frac{3}{4})^x\)</p>
<p>Graph A: \(q(x)=200 \cdot (\frac{7}{8})^x\)</p>
<ol class="os-raise-noindent" start="2">
<ol class="os-raise-noindent" type="a">
<li>Use the graphing tool or technology outside the course. On the same coordinate plane, sketch a graph that
represents the functions, \(f\) and \(g\). It may be helpful to use a different color for each function. </li>
</ol>
</ol>
<p><strong>Answer:</strong><br>
<img alt class="img-fluid atto_image_button_text-bottom" height="291" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/a4e9692d3b065fc38cf82ddc68059ee05805610c" width="300">
</p>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Which function is decaying more quickly? </li>
</ol>
</ol>
<p><strong>Answer:</strong><br>
\(f(x)\)
</p>
<ol class="os-raise-noindent" type="a" start="3">
<li> Explain your reasoning regarding which function decays more quickly. </li>
</ol>
<p><strong>Answer:</strong> \(f(x)=1,000 \cdot (\frac{1}{10})^x\) has a smaller base than \(g(x)=1,000 \cdot
(\frac{9}{10})^x\). Both bases are less than 1, so both functions decrease, but \(f(x)=1,000 \cdot (\frac{1}{10})^x\)
decreases more rapidly. </p>
<h4>Activity Synthesis</h4>
<p>Focus the discussion on the connection between the numbers in the equation (especially the \(b\)) and the features of
the graph. Discuss questions such as:</p>
<ul>
<li>“In the first question, what does the point that the graphs share on the \(y\)-axis say about the
situation?” (All of the accounts started with $200.)</li>
<li>“How does Graph A compare to Graph D?” (Graph A appears more horizontal, so the function is decaying
at a slower rate.)</li>
<li>“What does the largest factor (\(\frac78\)) tell us? To which graph does it correspond?” (The function
loses only \(\frac18\) of its value when \(x\) increases by 1, so it is decaying the most slowly. Graph A must be
the graph of \(q(x)=200 \cdot (\frac{7}{8})^x\) since it shows the slowest decay.)</li>
<li>“What might a graph representing a function \(v\) given by \(v(x)=200 \cdot (\frac{99}{100})^x\) look
like?” (It would look very close to a horizontal line because nearly all of its value remains as \(x\)
increases by 1, so it is decaying very slowly.)</li>
<li>“What might a graph representing a function \(w\) given by \(w(x)=200 \cdot (\frac{1}{100})^x\) look
like?” (It will go toward 0 extremely quickly because it loses \(\frac{99}{100}\) of its value each time \(x\)
increases by 1, so it is decaying very quickly.)</li>
<li>Emphasize that, in situations characterized by exponential growth (when \(b>1\)), a larger value of \(b\) means
a curve that is more vertical. In situations characterized by exponential decay, where \(b\) is between 0 and 1, the
closer \(b\) is to 1, the more the graph approaches a horizontal line. Conversely, the smaller the value of \(b\),
the more swiftly it heads toward 0 (the more vertical the curve is) before it flattens out and approaches a
horizontal line.</li>
</ul>
<h3>5.12.3: 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 functions will decay the fastest?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(f(x)=0.86^x-10\)
</td>
<td>
Incorrect. Let’s try again a different way: This is actually decaying the slowest since 0.86 is closest
to 1. The answer is \(g(x)=0.34^x+2\).
</td>
</tr>
<tr>
<td>
\(g(x)=0.34^x+2\)
</td>
<td>
That’s correct! Check yourself: Of all of the functions, \(b=0.34\) is the closest to 0, so it is
decaying the fastest.
</td>
</tr>
<tr>
<td>
\(h(x)=1.34^x-90\)
</td>
<td>
Incorrect. Let’s try again a different way: This function is growing exponentially since \(b>1\).
The answer is \(g(x)=0.34^x+2\).
</td>
</tr>
<tr>
<td>
\(j(x)=0.5^x\)
</td>
<td>
Incorrect. Let’s try again a different way: This is decaying slower than 0.34. The answer is
\(g(x)=0.34^x+2\).
</td>
</tr>
</tbody>
</table>
<br>
<h3> 5.12.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>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.</em></p>
<h4>Matching Exponential Functions and Graphs</h4>
<p>For each of the following, match each function with one of the graphs in the picture below.</p>
<p><img alt="Graph of six exponential functions."
src="https://k12.openstax.org/contents/raise/resources/13b36ed8389344dc587cabc989318f563a0a1795" width="300"></p>
<p>Remember, the equation for an exponential function is \(y=a\cdot b^x\). The \(a\) will affect the \(y\)-intercept.
When \(b<1\), there is exponential decay. The closer \(b\) is to 0, the faster it decays. The closer \(b\) is to 1,
the slower it decays. When \(b>1\), the larger the value of \(b\), the faster it grows.</p>
<table class="os-raise-wideadjustedtable">
<thead>
<tr>
<th scope="col">Equation</th>
<th scope="col">\(y\)-intercept</th>
<th scope="col">Growth/Decay</th>
<th scope="col">Match</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<ol class="os-raise-noindent">
<li>\(f(x)=2(0.68)^x\) </li>
</ol>
</td>
<td>
\((0, 2)\)
</td>
<td>
Decay
</td>
<td>
B
</td>
</tr>
<tr>
<td>
<ol class="os-raise-noindent" start="2">
<li>\(
f(x)=2(1.28)^x\) </li>
</ol>
</td>
<td>
\((0, 2)\)
</td>
<td>
Growth
</td>
<td>
F
</td>
</tr>
<tr>
<td>
<ol class="os-raise-noindent" start="3">
<li>\(
f(x)=2(0.81)^x\) </li>
</ol>
</td>
<td>
\((0, 2)\)
</td>
<td>
Decay
</td>
<td>
A
</td>
</tr>
<tr>
<td>
<ol class="os-raise-noindent" start="4">
<li>\( f(x)=4(1.28)^x\)</li>
</ol>
</td>
<td>
\((0, 4)\)
</td>
<td>
Growth
</td>
<td>
D
</td>
</tr>
<tr>
<td>
<ol class="os-raise-noindent" start="5">
<li>\(f(x)=2(1.59)^x\)</li>
</ol>
</td>
<td>
\((0, 2)\)
</td>
<td>
Growth
</td>
<td>
E
</td>
</tr>
<tr>
<td>
<ol class="os-raise-noindent" start="6">
<li>\(f(x)=4(0.68)^x\)</li>
</ol>
</td>
<td>
\((0, 4)\)
</td>
<td>
Decay
</td>
<td>
C
</td>
</tr>
</tbody>
</table>
<br>
<h4> Try It: Matching Exponential Functions and Graphs</h4>
<p><img alt="Graph of six exponential functions." height="295"
src="https://k12.openstax.org/contents/raise/resources/a3588f5079a44777c05412324d2a3176b0b499d5" width="300"></p>
<p>In the graph above, which function, \(f(x)=a \cdot b^x\), has the largest value for \(a\)?</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 identify which has the largest value for \(a\):</p>
<p>In the function \(f(x)=a \cdot b^x\), \(a\) changes the \(y\)-intercept. The function with the largest
\(y\)-intercept is C. It crosses the \(y\)-axis at the highest point.</p>