a09f2f4c-5869-4baa-8441-d6d9868b59f5.html

<h4>Analyzing \(a\) and \(b\) in the Exponential Function \(y=ab^x\)</h4>
<p><img alt="Graph of six exponential functions." class="img-fluid atto_image_button_text-bottom" height="318" src="https://k12.openstax.org/contents/raise/resources/a3588f5079a44777c05412324d2a3176b0b499d5" width="300"></p>
<ol class="os-raise-noindent">
  <li> Which graph has the largest value for \(b\)? </li>
  <li> Which graph has the smallest value for \(b\)?</li>
  <li> Which graph has the largest value for \(a\)? </li>
  <li> Which graph has the smallest value for \(a\)? </li>
</ol>
<p>When using the form \(y=ab^x\), it is important to remember that \(a\) is the initial value, or the \(y\)-intercept, and \(b\) is the growth factor. The growth factor reflects an exponential growth function when the value is greater than one. The growth factor reflects an exponential decay function when the value is between zero and one.</p>
<p>For question 1, you are looking for the largest growth factor, which would mean an exponential growth function that is increasing the most rapidly. First, identify the graphs that reflect exponential growth, which would be D, E, and F, and then decide which is the steepest. Graph D increases much faster than both graphs E and F.</p>
<p>For question 2, you are looking for the steepest rate of exponential decay, so you need the graph that has the smallest \(b\)-value. The graphs that reflect exponential decay are graphs A, B, and C. The graph that has a \(b\)-value that is closest to 0 is graph C. Graph C has the steepest exponential decay because its \(b\)-value is the closest to 0 and, therefore, the smallest.
</p>
<p>For question 3, you are looking for the largest initial value or \(y\)-intercept. This could be with a growth or decay function; it just needs to cross the \(y\)-axis higher than the other graphs. Graph C has the highest \(a\) value.</p>
<p>For question 4, you are looking for the smallest initial value or \(y\)-intercept. This could be a growth or decay function; it just needs to cross the \(y\)-axis lower than the other graphs. Graph F has the lowest \(a\) value.</p>
<h4>Try It: Analyzing \(a\) and \(b\) in the Exponential Function \(y=ab^x\)</h4>
<p><img alt="Graph of six exponential functions." class="img-fluid atto_image_button_text-bottom" height="295" src="https://k12.openstax.org/contents/raise/resources/13b36ed8389344dc587cabc989318f563a0a1795" width="300"></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">
    <ol class="os-raise-noindent">
      <li> Which graphs represent exponential growth? </li>
      <li> Which graphs represent exponential decay? </li>
      <li> Which graph has the highest value for \(b\)? </li>
      <li> Which graph has the highest value for \(a\)? </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="Reveal1">
  <p>Compare your answers:</p>
  <p>Here is how to analyze and compare graphs of exponential functions:</p>
  <ol class="os-raise-noindent">
    <li> Graphs indicate exponential growth if they are increasing from left to right. Graphs D, E, and F represent exponential growth. </li>
    <li> Graphs indicate exponential decay if they are decreasing from left to right. Graphs A, B, and C represent exponential decay. </li>
    <li> Graph E has the highest value of \(b\), or growth rate, since it is increasing faster than graphs D and F. </li>
    <li> Graphs C and D have the highest value for \(a\) because they have the same \(y\)-intercept. </li>
  </ol>
</div>