48281e98-d3ac-4225-a6cd-e6f7925b02a0.html

<h4>Activity (20 minutes)</h4>
<p>This activity prompts students to look at how the slope in slope-intercept form affects the graph of a linear function. Circulate to ensure students are graphing their lines properly. If necessary, partner students during the graphing process.</p>
<p>In pairs, students should discuss the vertical shifts to the parent function, \(f(x)=x\), to graph the new functions.</p>
<h4> Launch</h4>
<p>Allow students to work in pairs to check each other&rsquo;s graphs and discuss the relationships to answer the questions. During class discussion, solicit a pair for each question to describe the effects the different slopes have on the graphs.</p>
<p>For students who can benefit from a more accessible model, use geoboards and rubber bands. Use one color rubber band for the \(x\)- and \(y\)-axes. Use a different color for each line graphed. Work with students to count spots on the geoboard and discuss the graphs and their relationships.</p>

<h4>Student Activity </h4>
<h4>Activity</h4>
<h4>Vertical Stretches and Compressions</h4>

<ol class="os-raise-noindent">
  <li>On the same coordinate grid graph the following linear functions. You do not need to label the lines in the graphing tool. <br> (Students were provided access to Desmos)</li>
</ol>

<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">Line Label</th>
      <th scope="col">Function</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Line \(a\)</td>
      <td>\(f(x)=3x\)</td>
    </tr>
    <tr>
      <td>Line \(b\)</td>
      <td>\(f(x)=2x\)</td>
    </tr>
    <tr>
      <td>Line \(c\)</td>
      <td>\(f(x)=x\)</td>
    </tr>
  </tbody>
</table>

<p><strong>Answer:</strong> Compare your work:<br><img src="https://k12.openstax.org/contents/raise/resources/c97311aa5d27133b264f0ccb2501a6982f17df94" height="300"></p>
<br>

<ol class="os-raise-noindent" start="2">
  <li>What do you notice about the slopes of lines \(c, b,\) and \(a\) as the number of the coefficient gets larger?</li>
</ol>
<p><strong>Answer:</strong> As the coefficient increases, the line gets steeper.</p>
<br>

<ol class="os-raise-noindent" start="3">
  <li>Now, graph lines d and e on the same graph as \(a-c\).</li>
</ol>

<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">Line Label</th>
      <th scope="col">Function</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Line \(d\)</td>
      <td>\(f(x)=\frac12 x\)</td>
    </tr>
    <tr>
      <td>Line \(e\)</td>
      <td>\(f(x)= \frac13 x\)</td>
    </tr>
  </tbody>
</table>

<p><strong>Answer:</strong> Compare your work:<br><img src="https://k12.openstax.org/contents/raise/resources/78cae49817f2c19a2881f83d7e686ffa6ce15084" height="300"></p>
<br>

<ol class="os-raise-noindent" start="4">
  <li>How does line \(d\), representing \(f(x) = \frac{1}{2} x\), compare to line \(b\), representing \(f(x) = 2x\)?</li>
</ol>
<p><strong>Answer:</strong> As the coefficient gets smaller, the line is less steep.</p>
<br>

<ol class="os-raise-noindent" start="5">
  <li>Examine lines \(d\) and \(e\). What can you say about the lines when \(0 \lt m \lt 1\)?</li>
</ol>
<p><strong>Answer:</strong> As the coefficient gets smaller, the line is less steep.</p>
<br>

<ol class="os-raise-noindent" start="6">
  <li>Now graph lines \(f, g,\) and \(h\) on the same coordinate grid.You do not need to label the lines in the graphing tool. <br> (Students were provided access to Desmos)</li>
</ol>

<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">Line Label</th>
      <th scope="col">Function</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Line \(f\)</td>
      <td>\(f(x)=-2x\)</td>
    </tr>
    <tr>
      <td>Line \(g\)</td>
      <td>\(f(x)= -x\)</td>
    </tr>
    <tr>
      <td>Line \(h\)</td>
      <td>\(f(x)= -\frac{1}{2} x\)</td>
    </tr>
  </tbody>
</table>

<p><strong>Answer:</strong> Compare your work:<br><img src="https://k12.openstax.org/contents/raise/resources/20082432ee9342f82f6e63670d36c30c90254a1f" height="300"></p>
<br>

<ol class="os-raise-noindent" start="7">
  <li>Compare the equations of lines \(f-h\) to the graphs of lines \(b-d\). What changed with the equations of these functions?</li>
</ol>
<p><strong>Answer:</strong> There is a negative coefficient in lines \(f-h\) while in lines \(b-d\), they are positive.</p>
<br>

<ol class="os-raise-noindent" start="8">
  <li>Compare the graphs of lines \(f-h\) to the graphs of lines \(b-d\). How did the change in their equations affect their graphs?</li>
</ol>
<p><strong>Answer:</strong> Since there is a negative coefficient, the graphs of lines \(f-h\) are reflections of the positive versions of the lines over the \(x\)-axis. The graphs were vertically “flipped.”</p>
<br>

<p>In the equation \(f(x)=ax\), the \(a\) is acting as the vertical stretch or compression of the parent function. It dilates the graph. When \(a\) is negative, there is also a vertical reflection of the graph.</p>

<ul>
  <li>Multiplying the equation of \(f(x)=x\) by \(a\) vertically stretches or expands the graph of \(f\) by a factor of \(a\) units if \(a > 1\).</li>
  <li>The \(a\) vertically compresses the graph of \(f\) by a factor of \(a\) units if \(0 \lt a \lt 1\).</li>
  <li>This means the larger the absolute value of \(a\), the steeper the slope.</li>
</ul>

<p><img src="https://k12.openstax.org/contents/raise/resources/c022489323343d3f67bbc9b025766d0cc7fbe1b2" width="300"></p>

<p>In \(f(x)=mx+b\), the \(m\) acts as the vertical dilation (stretch or compression factor). This is a type of transformation to the graph of \(f(x)=x\).</p>

<div class="os-raise-graybox">
  <h5>Vertical Dilation</h5>
  <p>A vertical stretch or compression “transforms” the parent function into another function by vertically dilating the graph by “a” units.</p>
  <p align="center"> Vertical Dilation &rarr; \( af(x) \)</p>
</div>
<br>
<p>If the \(a\) value is negative, the graph is reflected over the \(x\)-axis. If the \(a\) value is greater than 1, the graph is stretched by \(a\) scale factor of \(a\). If the a value is between 0 and 1, the graph is compressed by a scale factor of \(a\).</p>
<br>
<h4>Video: Identifying Vertical Stretches and Compressions</h4>
<p>Watch the following video to learn more about vertical stretches and compressions.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
  <div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
      <source src="https://k12.openstax.org/contents/raise/resources/ab2d0745a29cce32d6c39a82461fc1d15be37057">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/b8785f2e173176360bdf2f05e88af8014891800f" srclang="en_us">https://k12.openstax.org/contents/raise/resources/ab2d0745a29cce32d6c39a82461fc1d15be37057
    </video></div>
</div>
<br>
<h4>Activity Synthesis</h4>
<p>Explain to students that graphing linear functions using transformations will assist them as you move to graphing different function families in the future.</p>
<h3>4.11.4: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</em></p>  
<p class="os-raise-text-bold">QUESTION:</p>
<p>How does the graph of \(f(x)=\frac14 x -3\) change from the parent function, \(f(x)=x\)?</p>
<table class="os-raise-textheavytable">
   <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
 </thead>
  <tbody>
    <tr> 
      <td>
        <p>Vertical stretch of \(\frac14\), down 3</p>
      </td>
      <td>
        <p>Incorrect. Let&rsquo;s try again a different way: A slope less than 1 signifies a vertical compression. The answer is vertical compression of \(\frac14\), down 3.</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>Vertical compression of \(\frac14\), down 3</p>
      </td>
      <td>
        <p>That&rsquo;s correct! Check yourself: The slope is \(\frac14\), which is less than 1, so there is a vertical compression. The \(y\)-intercept is &ndash;3, so there is a vertical shift down 3.</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>Vertical compression of \(\frac14\), up 3</p>
      </td>
      <td>
        <p>Incorrect. Let&rsquo;s try again a different way: The \(y\)-intercept is negative. The answer is vertical compression of \(\frac14\), down 3.</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>Vertical stretch of \(\frac14\), left 3</p>
      </td>
      <td>
        <p>Incorrect. Let&rsquo;s try again a different way: A slope less than 1 signifies a vertical compression, and the \(y\)-intercept gives the vertical shift. The answer is vertical compression of \(\frac14\), down 3.</p>
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>4.11.4: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it.</em></p>
<h4>Vertical Transformations of Linear Functions</h4>
<p>Given the equation of a linear function, use transformations to graph the linear function in the form \(f(x)=mx+b\).</p>
<p><strong>Example</strong></p>
<p>Graph \(f(x)= \frac12 x&minus;3\) using transformations.</p>
<p><strong>Solution:</strong></p>
<p><strong>Step 1 - </strong> Graph the parent function, \(f(x)=x\). </p>
<p><strong>Step 2 - </strong> Vertically stretch or compress the graph by a factor \(m\). The equation for the function shows that \(m=\frac12\), so the identity function is vertically compressed by \(\frac12\). </p>
<p><img alt="This graph shows two functions on an x, y coordinate plane. One shows an increasing function of y = x divided by 2 that runs through the points (0, 0) and (2, 1). The second shows an increasing function of y = x and runs through the points (0, 0) and (1, 1)." src="https://k12.openstax.org/contents/raise/resources/d5621b67f6db03da1c64ec9b78f685850687e2a2"></p>
<p>This graph shows two functions on an \((x, y)\) coordinate plane. One shows an increasing function of \(y = x\) divided by 2 that runs through the points \((0, 0)\) and \((2, 1)\). The second shows an increasing function of \(y = x\) and runs through the points \((0, 0)\) and \((1, 1)\).</p>
<p><strong>Step 3 - </strong> Shift the graph up or down \(b\) units.  </p>
<p> The equation for the function also shows that \(b=&minus;3\), so the parent function is vertically shifted down 3 units.</p>

<p>Now, show the vertical shift:</p>
<p><img alt="This graph shows two functions on an x, y coordinate plane. The first is an increasing function of y = x divided by 2 and runs through the points (0, 0) and (2, 1).  The second shows an increasing function of y = x divided by 2 minus 3 and passes through the points (0, 3) and (2, -2).  An arrow pointing downward from the first function  to the second function reveals the vertical shift." src="https://k12.openstax.org/contents/raise/resources/fa6330d94e6f7cd4b4fcc0bd7d135a06f1c34dd2"></p>
<p>This graph shows two functions on an \((x, y)\) coordinate plane. The first is an increasing function of \(y = x\) divided by 2 and runs through the points \((0, 0)\) and \((2, 1)\). The second shows an increasing function of \(y = x\) divided by 2 minus 3 and passes through the points \((0, 3)\) and \((2, -2)\). An arrow pointing downward from the first function to the second function reveals the vertical shift.</p>
<br>
<h4>Try It: Vertical Transformations of Linear Functions</h4>
<p>Name the transformations to go from the parent function, \(f(x)=x\), to \(f(x)=4+2x\).</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong></p>
<p>Here is how to determine the transformations:</p>
<p>First, the slope, \(m=2\), tells that there is a vertical stretch of 2.</p>
<p>Next, the \(y\)-intercept is 4, so there is a vertical shift up 4.</p>