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<h4>Activity</h4>
<p>Here is a table and a graph that show the number of coffee shops worldwide that a company had in its first 10 years,
between 1987 and 1997. The growth in the number of shops was roughly exponential.</p>
<p><img alt="Graph of a roughly exponential function, origin O. Years since 1987 and number of coffee shops." src="https://k12.openstax.org/contents/raise/resources/bdfc755f5f75d82877c744868cc66af19e924b2e"></p>
<br>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">Year</th>
<th scope="col">Number Of Shops</th>
</tr>
</thead>
<tbody>
<tr>
<td>
1987
</td>
<td>
17
</td>
</tr>
<tr>
<td>
1988
</td>
<td>
33
</td>
</tr>
<tr>
<td>
1989
</td>
<td>
55
</td>
</tr>
<tr>
<td>
1990
</td>
<td>
84
</td>
</tr>
<tr>
<td>
1991
</td>
<td>
116
</td>
</tr>
<tr>
<td>
1992
</td>
<td>
165
</td>
</tr>
<tr>
<td>
1993
</td>
<td>
272
</td>
</tr>
<tr>
<td>
1994
</td>
<td>
425
</td>
</tr>
<tr>
<td>
1995
</td>
<td>
677
</td>
</tr>
<tr>
<td>
1996
</td>
<td>
1,015
</td>
</tr>
<tr>
<td>
1997
</td>
<td>
1,412
</td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent">
<li> Find the average rate of change for each period of time. Show your reasoning.
</li>
</ol>
<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">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li>1987 to 1990 </li>
</ol>
</ol>
</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 answer:</p>
<p>\(22\frac13\) stores per year since \(\frac{(84−17)}{(3−0)}=22 \frac13\). </p>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> 1987 to 1993 </li>
</ol>
</ol>
</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="Reveal2">
<p>Compare your answer:</p>
<p> 42.5 stores per year since \(\frac{(272−17)}{(6−0)}=42.5\).</p>
</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">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li>1987 to 1997 </li>
</ol>
</ol>
</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 answer:</p>
<p>139.5 stores per year since \(\frac{(1412−17)}{(10−0)}=139.5\). </p>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal4" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>Make some observations about the rates of change you calculated. What do these average rates tell us about how
the company was growing during this time period?</li>
</ol>
</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="Reveal4">
<p>Compare your answer:</p>
<p>The average rate of change of the first 6 years is almost double that of the first 3 years. The average rate of
change of the 10-year period is almost 7 times greater than that of the first 3 years and more than 3 times greater
than the first 6 years. The company was growing at an increasing rate over the 10-year time period.</p>
</div>
<ol class="os-raise-noindent" start="3">
<li>Use the graph to support your answers to these questions. How well do the average rates of change describe the
growth of the company in:</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">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li>the first 3 years?</li>
</ol>
</ol>
</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 answer:</p>
<p>An average rate of change of \(22\frac{1}{3}\) is a fairly accurate prediction of growth over the first 3 years,
but it is a poor prediction of growth much beyond 1992.</p>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal6" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>the first 6 years?</li>
</ol>
</ol>
</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="Reveal6">
<p>Compare your answer:</p>
<p>An average rate of change of 42.5 is an okay prediction of growth over the first 6 years, but it overestimates
early on and greatly underestimates the growth in years after 1994.</p>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal7" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li>the entire 10 years?</li>
</ol>
</ol>
</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="Reveal7">
<p>Compare your answer:</p>
<p>An average rate of change of 139.5 does not accurately predict the actual growth of the company between 1987 and
1997.</p>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal8" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="4">
<li>Let \(f\) be the function so that \(f(t)\) represents the number of stores \(t\) years since 1987. The
value of \(f(20)\) is 15,011. Find \(\frac{f(20)−f(10)}{20−10}\) and say what it tells us about the
change in the number of stores.</li>
</ol>
</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="Reveal8">
<p>Compare your answer:</p>
<p>The average rate of growth from 1997 to 2007 is 1,359.9 stores per year. This is 60 times greater than the average
rate of change over the first 3 years and almost 10 times greater than the average rate of change over the first 10
years of the time period. The number of stores continued to grow at an increasing rate from 1997 to 2007.</p>
</div>
<br>
<h4>Video: Analyzing Average Rate of Change of an Exponential Function</h4>
<p>Watch the following video to learn more about the average rate of change of exponential 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/f012f2931d956078be6f22f90b24f63f50921333">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/f92dfb251ff3725fb06898b80dae14782cba7df4" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/f012f2931d956078be6f22f90b24f63f50921333
</video></div>
</div>