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<h4>Identifying the Constant Ratio of the Exponential Function</h4>
<p>What is the constant ratio (also called the growth factor) of the exponential function shown in the table? Rewrite each term to show the initial value and repeated use of the constant ratio.</p>
<table class="os-raise-horizontaltable">
<thead>
</thead>
<tbody>
<tr>
<th scope="row"> \(x\) </th>
<td> 0 </td>
<td> 1 </td>
<td> 2 </td>
<td> 3 </td>
<td> 4 </td>
</tr>
<tr>
<th scope="row"> \(f(x)\) </th>
<td> 4 </td>
<td> 12 </td>
<td> 36 </td>
<td> 108 </td>
<td> 324 </td>
</tr>
</tbody>
</table>
<br>
<h4>Defining Exponential Growth</h4>
<p>Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.</p>
<br>
<br>
<div class="os-raise-graybox"> <p> <strong> EXPONENTIAL GROWTH </strong> </p>
<hr>
<p>A function that models <strong>exponential growth </strong>grows by a rate proportional to the amount present. For any real number \(x\) and any positive real numbers \(a\) and \(b\) such that \(b\neq1\), an exponential growth function has the form</p>
<p align="center">\(f(x)=ab^x\)</p>
<p>where</p>
<ul>
<li>\(a\) is the initial or starting value of the function.</li>
<li>\(b\) is the growth factor or growth multiplier per unit \(x\).</li>
</ul>
</div>
<br>
<br>
<p>The initial or starting value of the function is 4, because that is when the value of \(x\) is 0.</p>
<p>The constant ratio, \(b\), would be 3, since every value in the table is 3 times the previous term.</p>
<table class="os-raise-horizontaltable">
<thead>
</thead>
<tbody>
<tr>
<th scope="row"> \(x\) </th>
<td> 0 </td>
<td> 1 </td>
<td> 2 </td>
<td> 3 </td>
<td> 4 </td>
</tr>
<tr>
<th scope="row"> \(f(x)\) </th>
<td> 4 </td>
<td> \(4\cdot3^1\) </td>
<td> \(4\cdot3^2\) </td>
<td> \(4\cdot3^3\) </td>
<td> \(4\cdot3^4\) </td>
</tr>
</tbody>
</table>
<br>
<h4>Try It: Identifying the Constant Ratio of the Exponential Function</h4>
<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">
<p>What is the constant ratio of the following exponential function?</p>
<table class="os-raise-horizontaltable">
<thead>
</thead>
<tbody>
<tr>
<th scope="row"> \(x\) </th>
<td> 1 </td>
<td> 2 </td>
<td> 3 </td>
<td> 4 </td>
</tr>
<tr>
<th scope="row"> \(f(x)\) </th>
<td> 10 </td>
<td> 20 </td>
<td> 40 </td>
<td> 80 </td>
</tr>
</tbody>
</table>
<br>
</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>Here is how to identify the exponential constant ratio of a table:</p>
<p>First, make sure the table is reflecting an exponential relationship by noticing how quickly the terms grow. Then, if the \(x\) values are increasing by 1, divide consecutive terms to determine the growth factor.</p>
<p>Since \(20\div 10=2\), \(40\div 20=2\), and \(80\div 40=2\), the growth factor is 2.</p>
</div>
<br>