ac4d5212-68b4-44d6-ab00-f510f7d53760.html

<h4> Determining the Growth Rate of Exponential Functions</h4>
<p>For the table below, assume the function \(f\) is defined for all real numbers. Calculate \(\triangle f =
  f(x+1)-f(x)\) in the last column. (The symbol \(\triangle\) in this context means &ldquo;change in.&rdquo;) What do
  you notice about \(\triangle f\)? Could the function be linear or exponential? Write a linear or an exponential
  function formula that generates the same input&ndash;output pairs as given in the table.</p>
<table class="os-raise-wideequaltable">
  <thead>
    <tr>
      <th scope="col">
        \(x\)
      </th>
      <th scope="col">
        \(f(x)\)
      </th>
      <th scope="col">
        \(\triangle f=f(x+1)-f(x)\)
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        0
      </td>
      <td>
        2
      </td>
      <td></td>
    </tr>
    <tr>
      <td>
        1
      </td>
      <td>
        6
      </td>
      <td></td>
    </tr>
    <tr>
      <td>
        2
      </td>
      <td>
        18
      </td>
      <td></td>
    </tr>
    <tr>
      <td>
        3
      </td>
      <td>
        54
      </td>
      <td></td>
    </tr>
    <tr>
      <td>
        4
      </td>
      <td>
        162
      </td>
      <td></td>
    </tr>
  </tbody>
</table>
<br>
<p>For each row, you subtract \(f(x)\) values. The first row would be \(6- 2 =4\), then \(18 - 6 =12\), then \(54 - 18
  =36\), and then \(162 - 54 = 108\). Since the rate of change isn&rsquo;t consistent, it is not a linear
  function.&nbsp; Exponential functions have a growth factor or a repeated rate that terms are being multiplied by. If
  you divide consecutive terms, you will find the growth rate is 3. Notice that the change between consecutive terms
  also has a growth rate of 3.</p>
<p>Since this function is exponential, an equation will have the form \(f(x) = initial \;\ value \;\ (growth \;\
  rate)^x\). The growth rate is 3. The initial value is the same as the \(y\)-intercept, or where \(x=0\). \(f(0)\) is
  given in the table, so the initial value is 2. The equation for this table would be \(f(x) = 2(3)^x\).</p>
<h4>Try It: Determining the Growth Rate of Exponential Functions</h4>
<p>For the graph provided, assume that the function is defined for all real numbers. What is the rate of change? Write
  an equation that would define this function.</p>
<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><img
        alt="GRAPH OF AN EXPONENTIAL GROWTH FUNCTION WITH A \(y\)-intercepts OF 1 AND PASSING THROUGH THE POINTS 1 COMMA 2, 2 COMMA 4, AND 3 COMMA 8"
        class="img-fluid atto_image_button_text-bottom" height="298"
        src="https://k12.openstax.org/contents/raise/resources/6694afad3354ea7171a966c8e63234938158c194" width="300">
    </p>
  </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 write the equation of an exponential function from a graph:</p>
  <p>The graph seems to be exponential. If you find the rate of change between consecutive points, you find the graph is
    exponential and the rate of change is 2 since the \(y\)-values increase by a factor of 2 every time the \(x\)-values
    increase by 1. Since the graph crosses the \(y\)-axis at \((0,1)\), that is the initial value. The equation for this
    graph is \(f(x) = 2^x\).</p>
</div>