01e6cf7a-9fcd-4bc5-ba36-0dcaab7aab10.html

<p>Previously, we saw that an exponential function eventually overtakes a linear function. We can understand better why
  this is so by examining how these functions change when we change their input. Present these equations and tables for
  functions \(f\) and \(g\):</p>
<p>\(f(x)=3x+2\)</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>
      <td>
        5
      </td>
      <td>
        10
      </td>
      <td>
        15
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(f(x)\)
      </th>
      <td>
        2
      </td>
      <td>
        5
      </td>
      <td>
        8
      </td>
      <td>
        11
      </td>
      <td>
        14
      </td>
      <td>
        17
      </td>
      <td>
        32
      </td>
      <td>
        47
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>\(g(x)=2 \cdot 3^x\)</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>
      <td>
        5
      </td>
      <td>
        10
      </td>
      <td>
        15
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(g(x)\)
      </th>
      <td>
        2
      </td>
      <td>
        6
      </td>
      <td>
        18
      </td>
      <td>
        54
      </td>
      <td>
        162
      </td>
      <td>
        486
      </td>
      <td>
        118,098
      </td>
      <td>
        28,697,814
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>Ask students:</p>
<ul>
  <li>&ldquo;How does the function \(f\) change when \(x\)<em> </em>grows by 1? By 5? By 10?&rdquo;</li>
  <li>&ldquo;How does the function \(g\) change when \(x\) grows by 1? By 5? By 10?&rdquo;</li>
</ul>
<p>Make sure students see that any time \(x\) increases by 2, \(f\) grows by \(3 \cdot 2\) or 6,
  and&nbsp;\(g\)&nbsp;grows by a factor \(3^2\) or 9. Any time \(x\) increases by 5, the value of \(f\) grows by \(3
  \cdot 5\) or 15, but the value of&nbsp;\(g\)&nbsp;grows by a factor of \(3^5\) or 243.</p>