7779b9ac-0b06-42f7-8f60-023b0124d6fb.html

<h4>Activity (10 minutes)</h4>
<p>Earlier, students made a series of observations about how changing the parameters of quadratic expressions
  transformed the graphs. They may not have fully recognized why the graphs changed the way they did. This activity
  prompts students to explain their earlier observations and further understand the behaviors of the graphs in relation
  to the quadratic expressions.</p>
<p>Students evaluate quadratic expressions with different coefficients and constant terms and compare the values to
  those expressions to the values of \(x^2\). Upon studying the table of values, they see, for example, that adding a
  constant term increases the value of \(x^2\) by that number, moving the corresponding points on the graph up by that
  amount. They see that multiplying the squared term by 2 or 12 changes the output values by a factor of 2 or 12, which
  moves the points for \(x^2\) to a position twice or half as high on the graph.</p>
<p>Making a spreadsheet tool available gives students an opportunity to choose appropriate tools strategically.</p>
<h4>Launch</h4>
<p>If students have access to a spreadsheet, suggest that it might be a helpful tool in this activity to speed up the
  calculation process.</p>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
  <li>Use the table with different values of \(x\) to answer questions a&ndash;g for the values of \(x^2+10\). (You may
    also use a spreadsheet tool, if available.)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2+10\)
      </th>
      <td>
        a. ____
      </td>
      <td>
        b. ____
      </td>
      <td>
        c. ____
      </td>
      <td>
        d. ____
      </td>
      <td>
        e. ____
      </td>
      <td>
        f. ____
      </td>
      <td>
        g. ____
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" type="a">
  <li> When \(x=-3\) and \(x^2=9\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((-3)^2+10=9+10=19\)
  </li>
</ol>
<ol class="os-raise-noindent" start="2" type="a">
  <li> When \(x=-2\) and \(x^2=4\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((-2)^2+10=4+10=14\)
  </li>
</ol>
<ol class="os-raise-noindent" start="3" type="a">
  <li> When \(x=-1\) and \(x^2=1\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((-1)^2+10=1+10=11\)
  </li>
</ol>
<ol class="os-raise-noindent" start="4" type="a">
  <li> When \(x=0\) and \(x^2=0\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((0)^2+10=0+10=10\)
  </li>
</ol>
<ol class="os-raise-noindent" start="5" type="a">
  <li> When \(x=1\) and \(x^2=1\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((1)^2+10=1+10=11\)
  </li>
</ol>
<ol class="os-raise-noindent" start="6" type="a">
  <li> When \(x=2\) and \(x^2=4\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((2)^2+10=4+10=14\)
  </li>
</ol>
<ol class="os-raise-noindent" start="7" type="a">
  <li> When \(x=3\) and \(x^2=9\), what is value of \(x^2+10\)? <br>
    <br>
    <strong>Answer:</strong> \((3)^2+10=9+10=19\)
  </li>
</ol>
<ol class="os-raise-noindent" start="2">
  <li>Use the table with different values of \(x\) to answer questions a&ndash;g for the values of \(x^2-3\). (You may
    also use a spreadsheet tool, if available.)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2-3\)
      </th>
      <td>
        a. ____
      </td>
      <td>
        b. ____
      </td>
      <td>
        c. ____
      </td>
      <td>
        d. ____
      </td>
      <td>
        e. ____
      </td>
      <td>
        f. ____
      </td>
      <td>
        g. ____
      </td>
    </tr>
  </tbody>
</table>
<br>
<br>
<ol class="os-raise-noindent" type="a">
  <li> When \(x=-3\) and \(x^2=9\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer: </strong>\((-3)^2-3=9-3=6\)
  </li>
</ol>
<ol class="os-raise-noindent" start="2" type="a">
  <li> When \(x=-2\) and \(x^2=4\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer:</strong> \((-2)^2-3=4-3=1\)
  </li>
</ol>
<ol class="os-raise-noindent" start="3" type="a">
  <li> When \(x=-1\) and \(x^2=1\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer:</strong> \((-1)^2-3=1-3=-2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="4" type="a">
  <li> When \(x=0\) and \(x^2=0\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer: </strong>\((0)^2-3=0-3=-3\)
  </li>
</ol>
<ol class="os-raise-noindent" start="5" type="a">
  <li> When \(x=1\) and \(x^2=1\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer:</strong> \(-2. (1)2-3=1-3=-2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="6" type="a">
  <li> When \(x=2\) and \(x^2=4\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer:</strong> \((2)^2-3=4-3=1\)
  </li>
</ol>
<ol class="os-raise-noindent" start="7" type="a">
  <li> When \(x=3\) and \(x^2=9\), what is value of \(x^2-3\)? <br>
    <br>
    <strong>Answer: </strong>\((3)^2-3=9-3=6\)
  </li>
</ol>
<p>Select the <strong>solution</strong> button to see the solutions to questions 1 - 2 organized into a table.</p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2+10\)
      </th>
      <td>
        19
      </td>
      <td>
        14
      </td>
      <td>
        11
      </td>
      <td>
        10
      </td>
      <td>
        11
      </td>
      <td>
        14
      </td>
      <td>
        19
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2-3\)
      </th>
      <td>
        6
      </td>
      <td>
        1
      </td>
      <td>
        -2
      </td>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        1
      </td>
      <td>
        6
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" start="3">
  <li>In the earlier activity &ldquo;Transformations with Quadratic Functions,&rdquo; you observed the effects on the
    graph of adding or subtracting a constant term from \(x^2\). Study the values in the table created from questions 1
    and 2 in this activity. Use these values to explain why the graphs in the earlier activity changed the way they did
    when a constant term is added or subtracted.</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p>Answer: All values in the third row are 10 more than those in the second row, and all values in the last row are 3
  less than those in the second row. This suggests that all the points that represent \(x^2+10\) are 10 units higher on
  the graph than those for \(x^2\), and all the points that represent \(x^2-3\) are 3 units lower than those for
  \(x^2\), which explains why the graphs move up or down when a constant term is added or subtracted.</p>
<ol class="os-raise-noindent" start="4">
  <li>Use the table with different values of \(x\) to answer questions a&ndash;g for the values of \(2x^2\). (You may
  also use a spreadsheet tool, if available.)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(2x^2\)
      </th>
      <td>
        a. _____
      </td>
      <td>
        b. _____
      </td>
      <td>
        c. _____
      </td>
      <td>
        d. _____
      </td>
      <td>
        e. _____
      </td>
      <td>
        f. _____
      </td>
      <td>
        g. _____
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" type="a">
  <li> When \(x=-3\) and \(x^2=9\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(2(-3)^2=2 \cdot 9=18\)
  </li>
</ol>
<ol class="os-raise-noindent" start="2" type="a">
  <li> When \(x=-2\) and \(x^2=4\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer: </strong>\(2(-2)^2=2 \cdot 4=8\)
  </li>
</ol>
<ol class="os-raise-noindent" start="3" type="a">
  <li> When \(x=-1\) and \(x^2=1\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(2(-1)2=2 \cdot 1=2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="4" type="a">
  <li> When \(x=0\) and \(x^2=0\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(2(0)^2=2 \cdot 0=0\)
  </li>
</ol>
<ol class="os-raise-noindent" start="5" type="a">
  <li> When \(x=1\) and \(x^2=1\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(2(1)^2=2 \cdot 1=2\).
  </li>
</ol>
<ol class="os-raise-noindent" start="6" type="a">
  <li> When \(x=2\) and \(x^2=4\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(2(2)^2=2 \cdot 4=8\)
  </li>
</ol>
<ol class="os-raise-noindent" start="7" type="a">
  <li> When \(x=3\) and \(x^2=9\), what is value of \(2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(2(3)^2=2 \cdot 9=18\)
  </li>
</ol>
<ol class="os-raise-noindent" start="5">
  <li>Use the table with different values of \(x\) to answer questions a&ndash;g for the values of \(\frac{1}{2}x^2\.
  (You may also use a spreadsheet tool, if available.)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(\frac{1}{2}x^2\)
      </th>
      <td>
        a. _____
      </td>
      <td>
        b. _____
      </td>
      <td>
        c. _____
      </td>
      <td>
        d. _____
      </td>
      <td>
        e. _____
      </td>
      <td>
        f. _____
      </td>
      <td>
        g. _____
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" type="a">
  <li> When \(x=-3\) and \(x^2=9\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(\frac{1}{2}(-3)^2=\frac{1}{2} \cdot 9=\frac{9}{2}=4\frac{1}{2}=4.5\)
  </li>
</ol>
<ol class="os-raise-noindent" start="2" type="a">
  <li> When \(x=-2\) and \(x^2=4\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(\frac{1}{2}(-2)^2=\frac{1}{2} \cdot 4=\frac{4}{2}=2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="3" type="a">
  <li> When \(x=-1\) and \(x^2=1\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(\frac{1}{2}(-1)^2=\frac{1}{2} \cdot 1=\frac{1}{2}=0.5\)
  </li>
</ol>
<ol class="os-raise-noindent" start="4" type="a">
  <li> When \(x=0\) and \(x^2=0\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(\frac{1}{2}(0)^2=\frac{1}{2} \cdot 0=0\)
  </li>
</ol>
<ol class="os-raise-noindent" start="5" type="a">
  <li> When \(x=1\) and \(x^2=1\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(\frac{1}{2}(1)^2=\frac{1}{2} \cdot 1=\frac{1}{2}=0.5\)
  </li>
</ol>
<ol class="os-raise-noindent" start="6" type="a">
  <li> When \(x=2\) and \(x^2=4\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer: </strong>\(\frac{1}{2}(2)2=\frac{1}{2} \cdot 4=\frac{4}{2}=2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="7" type="a">
  <li> When \(x=3\) and \(x^2=9\), what is value of \(\frac{1}{2}x^2\)? <br>
    <br>
    <strong>Answer: </strong>\(\frac{1}{2}(3)^2=\frac{1}{2} \cdot 9=\frac{9}{2}=4\frac{1}{2}=4.5\)
  </li>
</ol>
<ol class="os-raise-noindent" start="6">
  <li>Use the table with different values of \(x\) to answer questions a&ndash;g for the values of \(-2x^2\) . (You may
  also use a spreadsheet tool, if available.)</li>
</ol>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
        <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
 
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(-2x^2\)
      </th>
      <td>
        a. _____
      </td>
      <td>
        b. _____
      </td>
      <td>
        c. _____
      </td>
      <td>
        d. _____
      </td>
      <td>
        e. _____
      </td>
      <td>
        f. _____
      </td>
      <td>
        g. _____
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" type="a">
  <li> When \(x=-3\) and \(x^2=9\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(-3)^2=-2 \cdot 9=-18\)
  </li>
</ol>
<ol class="os-raise-noindent" start="2" type="a">
  <li> When \(x=-2\) and \(x^2=4\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(-2)^2=-2 \cdot 4=-8\)
  </li>
</ol>
<ol class="os-raise-noindent" start="3" type="a">
  <li> When \(x=-1\) and \(x^2=1\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(-1)^2=-2 \cdot 1=-2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="4" type="a">
  <li> When \(x=0\) and \(x^2=0\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(0)^2=-2 \cdot 0=0\)
  </li>
</ol>
<ol class="os-raise-noindent" start="5" type="a">
  <li> When \(x=1\) and \(x^2=1\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(1)^2=-2 \cdot 1=-2\)
  </li>
</ol>
<ol class="os-raise-noindent" start="6" type="a">
  <li> When \(x=2\) and \(x^2=4\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(2)^2=-2 \cdot 4=-8\)
  </li>
</ol>
<ol class="os-raise-noindent" start="7" type="a">
  <li> When \(x=3\) and \(x^2=9\), what is value of \(-2x^2\)? <br>
    <br>
    <strong>Answer:</strong> \(-2(3)^2=-2 \cdot 9=-18\)
  </li>
</ol>
<p>Select the <strong>solution</strong> button to see the solutions to questions 4&ndash;6 organized into a table.</p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(2x^2\)
      </th>
      <td>
        18
      </td>
      <td>
        8
      </td>
      <td>
        2
      </td>
      <td>
        0
      </td>
      <td>
        2
      </td>
      <td>
        8
      </td>
      <td>
        18
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(\frac{1}{2}x^2\)
      </th>
      <td>
        \(4\frac{1}{2}=4.5\)
      </td>
      <td>
        2
      </td>
      <td>
        \(\frac{1}{2}=0.5\)
      </td>
      <td>
        0
      </td>
      <td>
        \(\frac{1}{2}=0.5\)
      </td>
      <td>
        2
      </td>
      <td>
        \(4\frac{1}{2}=4.5\)
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(-2x^2\)
      </th>
      <td>
        -18
      </td>
      <td>
        -8
      </td>
      <td>
        -2
      </td>
      <td>
        0
      </td>
      <td>
        -2
      </td>
      <td>
        -8
      </td>
      <td>
        -18
      </td>
    </tr>
  </tbody>
</table>
<br>
<ol class="os-raise-noindent" start="7">
  <li>In the earlier activity &ldquo;Transformations with Quadratic Functions,&rdquo; you observed the effects on the
    graph of multiplying \(x^2\) by different coefficients. Study the values in the table created from questions 1 - 2
    in this activity. Use these values to explain why the graphs in the earlier activity changed the way they did when
    \(x^2\) is multiplied by a number greater than 1, by a negative number less than or equal to -1, and by numbers
    between -1 and 1.</li>
</ol>
<p>Write down your answer, then select the <strong>solution </strong>button to compare your work.</p>
<p><strong>Answer: </strong>When we multiply \(x^2\) by the positive coefficient 2, all the output values double. For
  any input value \(x\) , the output value of \(2x^2\) is twice as large as that of \(x^2\). So, the width of the graph
  becomes narrower because it grows faster. When we multiply \(x^2\) by a number between 0 and 1, the output values
  become smaller, so the graph becomes wider or vertically compressed. When we multiply by a negative number, all the
  positive values become negative, which reflects the direction of the opening of the graph from opening up to opening
  down.</p>
<h4>Video: Understanding the Behaviors of a Graph in Relation to Its Quadratic Expression</h4>
<p>Watch the following video to learn more about the behaviors of a graph in relation to its quadratic expression.</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/51a23e0458b71ce1bf5fda19ecbeb56d402c37cf">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/6490961d643c5011adcc5ed8c591599bb1aaec5e" srclang="en_us">
      https://k12.openstax.org/contents/raise/resources/51a23e0458b71ce1bf5fda19ecbeb56d402c37cf
    </video></div>
</div>
<br>
<br>
<h4>Activity Synthesis</h4>
<p>Invite students to share their analyses on how the values in the tables relate the behaviors of the graphs they saw
  earlier. Consider plotting the points on a dynamic graphing tool to make explicit the connections between the values
  in the tables and the graphs.</p>
<p><img alt="3 graphs of different colored plotted points on coordinate plane." height="262" src="https://k12.openstax.org/contents/raise/resources/6ac2d1b7327d2e2004d68cc0b946c2b4a1a1b6e2" width="268"></p>
<br>
<h3>7.12.3: 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>Which curve is narrower than \(f(x)=x^2\)?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">
        Answers
      </th>
      <th scope="col">
        Feedback
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        \(f(x)=x^2+3\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This will move the graph of f(x)=x^2 up 3 (a vertical shift).
        The answer is \(f(x)=3x^2\).
      </td>
    </tr>
    <tr>
      <td>
        \(f(x)=3x^2\)
      </td>
      <td>
        That&rsquo;s correct! Check yourself: Since the coefficient of \(f(x)=x^2\) is 3 and \(3&gt;1\), this will make
        the graph narrower or skinnier than the parent graph \(f(x)=x^2\) (a vertical stretch).
      </td>
    </tr>
    <tr>
      <td>
        \(f(x)=\frac{1}{3}x^2\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This will make the graph of \(f(x)=x^2\) wider (a vertical
        compression). The answer is \(f(x)=3x^2\).
      </td>
    </tr>
    <tr>
      <td>
        \(f(x)=(x+3)^2\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This will move the graph of \(f(x)=x^2\) to the left 3 (a
        horizontal shift). The answer is \(f(x)=3x^2\).
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>7.12.3: 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> Comparing Transformations to Quadratic Graphs</h4>
<p>Remember that the graph representing any quadratic function is a shape called a parabola. People often say that a
  parabola "opens upward" when the lowest point on the graph is the vertex (where the graph changes direction), and
  "opens downward" when the highest point on the graph is the vertex. Each coefficient in a quadratic expression written
  in standard form \(ax^2+bx+c\) tells us something important about the graph that represents it.</p>
<p>The graph of \(f(x)=x^2\) is a parabola opening upward with vertex at \((0,0)\). Adding a constant term 5 gives
  \(f(x)=x^2+5\) and raises the graph by 5 units. Subtracting 4 from \(x^2\) gives \(f(x)=x^2&minus;4\) and moves the graph 4
  units down.</p>
<p><img height="264" src="https://k12.openstax.org/contents/raise/resources/59c1850bb026bd2d08edf63adf0492da22758f93" width="264"></p>
<br>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2+5\)
      </th>
      <td>
        14
      </td>
      <td>
        9
      </td>
      <td>
        6
      </td>
      <td>
        5
      </td>
      <td>
        6
      </td>
      <td>
        9
      </td>
      <td>
        14
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2-4\)
      </th>
      <td>
        5
      </td>
      <td>
        0
      </td>
      <td>
        -3
      </td>
      <td>
        -4
      </td>
      <td>
        -3
      </td>
      <td>
        0
      </td>
      <td>
        5
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>A table of values can help us see that adding 5 to \(x^2\) increases all the output values of \(f(x)=x^2\) by 5, which
  explains why the graph moves up 5 units. Subtracting 4 from \(x^2\) decreases all the output values of \(f(x)=x^2\) by 4,
  which explains why the graph shifts down by 4 units.</p>
<br>
<p>In general, the constant term of a quadratic expression in standard form influences the vertical position of the
  graph. An expression with no constant term (such as \(x^2\) or \(x^2+9x\)) means that the constant term is 0, so the
  \(y\)-intercept of the graph is on the \(x\)-axis. It's not shifted up or down relative to the \(x\)-axis.</p>
<br>
<h4>Coefficient of the Squared Term
</h4>
<p>The coefficient of the squared term in a quadratic function also tells us something about its graph. The coefficient
  of the squared term in \(f(x)=x^2\) is 1. Its graph is a parabola that opens upward.</p>
<br>
<p>Multiplying \(x^2\) by a number greater than 1 makes the graph steeper, so the parabola is narrower than that
  representing \(x^2\).</p>
<p>Multiplying \(x^2\) by a number less than 1 but greater than 0 makes the graph less steep, so the parabola is wider
  than that representing \(x^2\).</p>
<p>Multiplying \(x^2\) by a number less than 0 makes the parabola open downward.</p>
<p><img height="264" src="https://k12.openstax.org/contents/raise/resources/92855a54cc0fbed5499158a46a879c93bb43788f" width="264"></p>
<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
      <th scope="row">
        \(x\)
      </th>
      <td>
        -3
      </td>
      <td>
        -2
      </td>
      <td>
        -1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        2
      </td>
      <td>
        3
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(x^2\)
      </th>
      <td>
        9
      </td>
      <td>
        4
      </td>
      <td>
        1
      </td>
      <td>
        0
      </td>
      <td>
        1
      </td>
      <td>
        4
      </td>
      <td>
        9
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(2x^2\)
      </th>
      <td>
        18
      </td>
      <td>
        8
      </td>
      <td>
        2
      </td>
      <td>
        0
      </td>
      <td>
        2
      </td>
      <td>
        8
      </td>
      <td>
        18
      </td>
    </tr>
    <tr>
      <th scope="row">
        \(-2x^2\)
      </th>
      <td>
        -18
      </td>
      <td>
        -8
      </td>
      <td>
        -2
      </td>
      <td>
        0
      </td>
      <td>
        -2
      </td>
      <td>
        -8
      </td>
      <td>
        -18
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>If we compare the output values of \(2x^2\) and \(-2x^2\), we see that they are opposites, which suggests that one
  graph would be a reflection of the other across the \(x\)-axis.</p>
<p>In the previous examples, parameters were being applied to the output values of the function. In other words, after an input value was squared, something was then added, subtracted, or multiplied with the squared value. The resulting impact on the graph was to vertically alter it.</p>

<p>However, what happens when the input value is altered before applying the exponent? In these situations, the impact on the graph is to produce a horizontal transformation.</p>

<h4>Horizontal Shifts</h4>

<p>Return to the quadratic parent function, \(f(x) = x^2\). The function is equivalent to \(f(x) = (x-0)^2\). While a bit obvious, this implies there is no change to the input value before being squared.</p>

<p>If the number used to change the input is greater than 0, then the graph of the function is shifted to the right.</p>

<p>Conversely, if the number used to change the input is less than 0, the graph of the function is shifted to the left. Note the impact to the subtraction operation when negative values are subtracted from the input.</p>

<p><img src="https://k12.openstax.org/contents/raise/resources/2d290cf005a548aa643a3f2d3174a0013ba6991f" width="300"/></p>

<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
        <th scope="row">\(x\)</th>
        <td>-3</td>
        <td>-2</td>
        <td>-1</td>
        <td>0</td>
        <td>1</td>
        <td>2</td>
        <td>3</td>
    </tr>
      <tr>
        <th scope="row">\(x^2\)</th>
        <td>9</td>
        <td>4</td>
        <td>1</td>
        <td>0</td>
        <td>1</td>
        <td>4</td>
        <td>9</td>
      </tr>
      <tr>
        <th scope="row">\((x - 3)^2\)</th>
        <td>36</td>
        <td>25</td>
        <td>16</td>
        <td>9</td>
        <td>4</td>
        <td>1</td>
        <td>0</td>
      </tr>
      <tr>
        <th scope="row">(\(x - -3)^2 = (x + 3)^2\)</th>
        <td>0</td>
        <td>1</td>
        <td>4</td>
        <td>9</td>
        <td>16</td>
        <td>25</td>
        <td>36</td>
      </tr>
  </tbody>
</table>
<br>
<p>Notice in this table of values that the change to the input values takes place before squaring the term. The constant is directly affecting the \(x\)-value before applying the function rule.</p>

<p>Examine the vertex of each graph. When we compare the graphs of  \((x-3)^2\) and  \((x+3)^2\) to the graph of the parent function, we see the parabola has been shifted to the right 3 units and left 3 units, respectively. Thus, a horizontal shift in the graph results when a constant has been subtracted from the input value.</p>

<h3>Horizontal Dilations and Reflections</h3>
<p>Now we will investigate the impact of a coefficient on the input value.</p>

<p>Multiplying the input value by a value greater than 1 and then applying the exponent, results in a parabola that has been horizontally compressed.</p>

<p>Multiplying the input value by a value that is between 0 and 1, yields a parabola that has been horizontally stretched.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/4d0fb7287fdeb5204da3c8264aec7597c6c7a42e" width="300"/></p>


<table class="os-raise-horizontaltable">
  <thead></thead>
  <tbody>
    <tr>
        <th scope="row">\(x\)</th>
        <td>-3</td>
        <td>-2</td>
        <td>-1</td>
        <td>0</td>
        <td>1</td>
        <td>2</td>
        <td>3</td>
      </tr>
      <tr>
        <th scope="row">\(x^2\)</th>
        <td>9</td>
        <td>4</td>
        <td>1</td>
        <td>0</td>
        <td>1</td>
        <td>4</td>
        <td>9</td>
      </tr>
      <tr>
        <th scope="row">\((2x)^2\)</th>
        <td>36</td>
        <td>16</td>
        <td>4</td>
        <td>0</td>
        <td>4</td>
        <td>16</td>
        <td>36</td>
      </tr>
      <tr>
        <th scope="row">\((\frac12x)^2\)</th>
        <td>2.25</td>
        <td>1</td>
        <td>0.25</td>
        <td>0</td>
        <td>0.25</td>
        <td>1</td>
        <td>2.25</td>
      </tr>
      <tr>
        <th scope="row">\((-2x)^2\)</th>
        <td>36</td>
        <td>16</td>
        <td>4</td>
        <td>0</td>
        <td>4</td>
        <td>16</td>
        <td>36</td>
      </tr>
    </tbody>
  </table>
<br>

<p>If the input value is multiplied by a negative number, then the parabola is reflected over the \(y\)-axis. However, due to the symmetric nature of parabolas, this is difficult to discern visually.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/60e3fcdea6928ea9078381f7b859c298b6ea8f95" width="300"/></p>

<p>Multiplying the input value by a number changes the quantity that is squared. As a result, the graph of the equation is horizontally stretched or compressed. <br>
  <br>
</p>
<h4>Try It: Comparing Transformations to Quadratic Graphs</h4>
<ol class="os-raise-noindent">
  <li>If the red graph (in the middle) is the parent function, \(f(x)=x^2\), what are the equations of the blue graph
    (at the top) and green graph (at the bottom)?</li>
</ol>
<p><img alt="GRAPH OF THE PARENT QUADRATIC FUNCTION IN RED. GRAPH OF THE PARENT FUNCTION SHIFTED UP 4 UNITS IN BLUE. GRAPH OF THE PARENT FUNCTION SHIFTED DOWN 2 UNITS IN GREEN." class="img-fluid atto_image_button_text-bottom" height="222" src="https://k12.openstax.org/contents/raise/resources/c972ea8c8fe74654651f1b82953adc656cafca94" width="223"></p>
<p>Write down your answer, then select the<strong> solution </strong>button to compare your work.</p>
<p><strong>Answer:</strong> Here is how to solve to find the equations of the other graphs:<br>
  <br>
  The green graph is the same as \(f(x)=x^2\) but is shifted down 2. The function will be \(f(x)=x^2-2\).<br>
  The blue graph is the same as \(f(x)=x^2\) but is shifted up 4. The function will be \(f(x)=x^2+4\).
</p>
<ol class="os-raise-noindent" start="2">
  <li>If the red graph (in the middle) is the parent function, \(f(x)=x^2\), what are the equations of the blue graph
    (to the right) and green graph (to the left)?</li>
</ol>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="277" role="presentation" src="https://k12.openstax.org/contents/raise/resources/3586113e777c12bb9b26bd811255476329e14e6c" width="278"></p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>Here is how to solve to find the equations of the other graphs.</p>
<p>The green graph is the same as \(f(x)=x^2\) but is shifted left \(\frac32\). The function will be
  \(f(x)=(x+\frac{3}{2})^2\).</p>
<p>The blue graph is the same as \(f(x)=x^2\) but is shifted right 3. The function will be \(f(x)=(x-3)^2\).</p>
<ol class="os-raise-noindent" start="3">
  <li>The red graph (in the middle) is\(f(x)=x^2\). The equations of the other graphs
  are\(f(x)=\frac{1}{3}x^2\) and \(f(x)=5x^2\).</li>
</ol>
<p>Which function matches the green graph (on the inside)? The blue graph (on the outside)?</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="335" role="presentation" src="https://k12.openstax.org/contents/raise/resources/224a8036304dedefb7ea91f1d9a6193a5b062eb1" width="340"></p>
<p>Write down your answer, then select the <strong>solution </strong>button to compare your work.</p>
<p><strong>Answer:</strong> The green graph is narrower than \(f(x)=x^2\). It must have a coefficient that is larger
  than the coefficient of \x^2\). The green graph is \(f(x)=5x^2\).</p>
<p>The blue graph is wider than \(f(x)=x^2\). It must have a coefficient that is smaller than the coefficient of
\(x^2\). The blue graph is \(f(x)=\frac{1}{3}x^2\).</p>
<ol class="os-raise-noindent" start="4">
  <li>The black graph (in the middle) is the parent function f(x) = x2. The equations of the other graphs are f(x) = (25x)2 and f(x) = (43x)2. 
    Which function matches the green graph (on the inside)? The blue graph (on the outside)?  
</li>
</ol>
<p><img src="https://k12.openstax.org/contents/raise/resources/59487ad1ef62dff2b803d636037399431c61b0f4" width="300"/></p>
<p>Write down your answer, then select the <strong>solution </strong>button to compare your work.&nbsp;</p>

<h5>Solution</h5>
<p>Here is how to solve to find the equations of the other graphs:</p>
<p>The green graph has been horizontally compressed in comparison to the parent function. Therefore, the coefficient impacting the input value must be greater than 1. That means the green graph represents \(f(x) = (\frac43x)^2\).</p>
<p>The blue graph has been horizontally stretched in comparison to the parent function. So, the coefficient impacting the input value must be greater than 0 but less than 1. That means the blue graph represents \(f(x) = (\frac25x)^2\).</p>