19e2cc9c-dd28-4788-abef-02a0bff5d004.html

<h4>Exponential versus Quadratic Functions</h4>
<p>For each of the two functions, do the following:</p>
<ol class="os-raise-noindent" type="a">
  <li> Complete the table of values. </li>
  <li> Sketch a graph. </li>
  <li> Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know. </li>
</ol>
<ol class="os-raise-noindent">
  <li> \(f(x)=3\cdot 2^x\)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead> </thead>
  <tbody>
    <tr>
      <th scope="row">\(x\)</th>
      <td>\(-1\)</td>
      <td>\(0\)</td>
      <td>\(1\)</td>
      <td>\(2\)</td>
      <td>\(3\)</td>
      <td>\(5\)</td>
    </tr>


    <tr>
      <th scope="row">\(f(x)\)</th>
      <td>\(\frac{3}{2}\) or equivalent </td>
      <td>\(3\)</td>
      <td>\(6\)</td>
      <td>\(12\)</td>
      <td>\(24\)</td>
      <td>\(96\)</td>
    </tr>
  </tbody>
</table>
<br>
<p><img height="298" src="https://k12.openstax.org/contents/raise/resources/88922da47465d7b4e2be357a2e6d1c8b87bb093b"
    width="300"></p>
<p>This function is exponential. It has a growth factor of \(2\).</p>
<ol class="os-raise-noindent" start="2">
  <li>\(g(x)=3\cdot x^2\)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">\(x\)</th>
      <td>\(-1\)</td>
      <td>\(0\)</td>
      <td>\(1\)</td>
      <td>\(2\)</td>
      <td>\(3\)</td>
      <td>\(5\)</td>
    </tr>
    <tr>
      <th scope="row">\(g(x)\)</th>
      <td>\(3\) </td>
      <td>\(0\)</td>
      <td>\(3\)</td>
      <td>\(12\)</td>
      <td>\(27\)</td>
      <td>\(75\)</td>
    </tr>
  </tbody>
</table>
<br>
<p><img height="304" src="https://k12.openstax.org/contents/raise/resources/c9c483e847a71d346d069f57164ded7e77f81809"
    width="300"></p>
<p>This function is not linear because it does not have a constant rate of change, and it is not exponential because it
  does not have a constant growth factor. The function is quadratic because it can be represented by \(g(x)=3x^2\),
  which has \(x\) to the second power, and that is the greatest power.</p>
<p>Note: Between \(f(x)\) and \(g(x)\), \(f(x)\) will have the greater value as \(x\) increases.</p>
<h4>Try It: Exponential versus Quadratic Functions</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>Which of the following has the greater value as \(x\) increases, \(f(x)\) or \(g(x)\)?</p>
    <p>\(f(x)=3x^2\)</p>
    <p>\(g(x)=3^x\)</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>Here is how to determine which function has the greater value as \(x\) increases:</p>
  <p>Make a table of values for each function, and then graph the function:</p>
  <table class="os-raise-horizontaltable">
    <thead></thead>
    <tbody>
      <tr>
        <th scope="row">\(x\)</th>
        <td>\(-2\)</td>
        <td>\(-1\)</td>
        <td>\(0\)</td>
        <td>\(1\)</td>
        <td>\(2\)</td>
        <td>\(3\)</td>
        <td>\(4\)</td>
      </tr>
      <tr>
        <th scope="row">\(f(x)=3x^2\)</th>
        <td>\(12\)</td>
        <td>\(3\)</td>
        <td>\(0\)</td>
        <td>\(3\)</td>
        <td>\(12\)</td>
        <td>\(27\)</td>
        <td>\(48\)</td>
      </tr>
    </tbody>
  </table>
  <br>
  <p><img alt height="304" role="presentation"
      src="https://k12.openstax.org/contents/raise/resources/f4631cb9d02c271990fc27002a5536212ecd6a16" width="300"></p>
  <table class="os-raise-horizontaltable">
    <thead></thead>
    <tbody>
      <tr>
        <th scope="row">\(x\)</th>
        <td>\(-2\)</td>
        <td>\(-1\)</td>
        <td>\(0\)</td>
        <td>\(1\)</td>
        <td>\(2\)</td>
        <td>\(3\)</td>
        <td>\(4\)</td>
      </tr>
      <tr>
        <th scope="row">\(g(x)=3^x\)</th>
        <td>\(\frac {1}{9}\)</td>
        <td>\(\frac {1}{3}\)</td>
        <td>\(1\)</td>
        <td>\(3\)</td>
        <td>\(9\)</td>
        <td>\(27\)</td>
        <td>\(81\)</td>
      </tr>
    </tbody>
  </table>
  <br>
  <p><img alt class="img-fluid atto_image_button_text-bottom" role="presentation"
      src="https://k12.openstax.org/contents/raise/resources/6ead768f506e99b9144290b0b041f973f9159381" width="260"></p>
  <p>Looking at both the table of values and the graphs, \(g(x)\) will have a greater value than \(f(x)\) as \(x\)
    increases. This is because \(g(x)\) is exponential and has a constant growth factor.</p>
</div>