ec4996f5-4140-45d7-b9c2-d7795b32a951.html

<h4>Determining Rate of Change of Linear 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>
        1
      </td>
      <td>
        -3
      </td>
      <td>
      </td>
    </tr>
    <tr>
      <td>
        2
      </td>
      <td>
        1
      </td>
      <td>
      </td>
    </tr>
    <tr>
      <td>
        3
      </td>
      <td>
        5
      </td>
      <td>
      </td>
    </tr>
    <tr>
      <td>
        4
      </td>
      <td>
        9
      </td>
      <td>
      </td>
    </tr>
    <tr>
      <td>
        5
      </td>
      <td>
        13
      </td>
      <td>
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>For each row, you subtract \(f(x)\) values. The first row would be \(1- (-3) =4\), then \(5 - 1 =4\), then \(9 - 5
  =4\), and then \(13 - 9 = 4\). Think back to the lessons on the slope or rate of change of a linear function. The
  numerator was always the change in \(y\) or output, which is the same as the change in \(f(x)\). Notice that all the
  changes in consecutive terms are the same, 4. This means that there is a constant rate of change and this particular
  function is linear.</p>
<p>Since this function is linear, an equation will have the form \(f(x) = rate \;\ of \;\ change \cdot x+ initial \;\
  value\). The rate of change is 4. The initial value is the same as the \(y\)-intercept, or where \(x=0\). \(f(0)\)
  would be 4 less than the first term listed in this case, which means that \(f(0)= -7\). The linear equation for this
  table would be \(f(x) = 4x -7\).</p>
<h4>Try It: Determining Rate of Change of Linear 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 A LINEAR FUNCTION WITH A \(y\)-intercepts OF 2 AND PASSING THROUGH THE POINTS 2 COMMA 3, 4 COMMA 4, 6 COMMA 5, AND 8 COMMA 6."
        class="img-fluid atto_image_button_text-bottom" height="606"
        src="https://k12.openstax.org/contents/raise/resources/9e3a19db72a3b8e7708a3d441ff551325ce9a03e" 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 the graph:</p>
  <p>The graph seems to be linear. If you find the rate of change between consecutive points, you find the graph is
    linear and the rate of change is \(\frac12\) since the \(y\)-values increase by 1 every time the \(x\)-values
    increase by 2. Since the graph crosses the \(y\)-axis at \((0,2)\), that is the initial value. The equation for this
    graph is \(f(x) = \frac{1}{2} x +2\).</p>
</div>