afb3f309-da04-4a26-9162-74d8c7525d3d.html

<h4>Activity (15 minutes)</h4>
<p>This activity enables students to see how the coefficient of the squared term and the constant term in a quadratic
  expression in standard form can be seen on the graph. Students start by graphing \(f(x)=x^2\) using technology. They then
  experiment with adding positive and negative constant values to \(x^2\) and multiplying it by positive and negative
  coefficients. They generalize their observations afterward. Along the way, students practice looking for and
  expressing regularity through repeated reasoning.</p>
<p>If working with a partner, students will take turns using the graphing technology and recording observations. As
  students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and
  critique the reasoning of others.</p>
<h4>Launch</h4>
<p>Provide access to devices that can run Desmos or other graphing technology. Consider arranging students in groups of
  two. Ask one partner to operate the graphing technology and the other to record the group's observations, and then to
  switch roles halfway.</p>
<p>If time is limited, consider arranging students in groups of four and asking each group member to experiment with one
  of the four changes listed in the activity and then reporting the results to their group.</p>
<p>Remind students to adjust their graphing window as needed. If using Desmos, instruct students that creating a slider
  might be a helpful tool for this activity. For students who need reminding, they can create sliders by typing a letter
  to represent a parameter, such as \(f(x)= x^2+a\), to change the constant term.</p>

<br>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for English Language Learners</p>
    <p class="os-raise-extrasupport-name">MLR 2 Discussion Supports: Representing, Conversing</p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>To support small-group discussion, invite students to share what they notice about how the graphs change or stay the same depending on how they change the function. Listen for and amplify language students use to describe the features of the graph such as “opens upward”, “opens downward”, “steeper”, “wider”, etc. Post the collected language in the front of the room so that students can refer to it throughout the rest of the activity and lesson.</p>
    <p class="os-raise-text-italicize">Design Principle(s): Maximize meta-awareness; Support sense-making</p>
    <p class="os-raise-extrasupport-title">Provide support for students</p>
      <p>
        <a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
           to the students to assist them with participating in this routine.
      </p>
  </div>
</div>
<br>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
    <p class="os-raise-extrasupport-name">Representation: Internalize Comprehension</p>
   </div>
  <div class="os-raise-extrasupport-body">
    <p>Represent the same information through different modalities by using individual sketches of each function. Provide students with a graphic organizer that provides space to include sketches and observations for each equation. Some students may benefit from additional support to learn what types of details are helpful in a sketch of a graph.</p>
    <p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing; Visual-spatial processing</p>
  </div>
</div>

<br>
<h4>Student Activity</h4>
<p>Use the graphing tool or technology outside the course. Graph \(f(x)=x^2\), and then experiment with each of the changes
  to the function in questions 1&ndash;7. Record your observations of what you noticed or wondered about each set of
  changes to the function (include sketches, if helpful).&nbsp;</p>
<ol class="os-raise-noindent">
  <li>Add different constant terms to \(x^2\) (for example: \(x^2+5\), \(x^2+10\), \(x^2-3\)).</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> Changing the constant term either raised the graph (when the constant is positive) or
  lowered the graph (when the constant is negative). Adding the constant term shifted the graph vertically, but the
  overall shape of the graph did not change.</p>
<ol class="os-raise-noindent" start="2">
  <li>Multiply \(x^2\) by different positive coefficients greater than 1 (for example: \(3x^2\), \(7.5x^2\)).</li>
</ol>
<p>Write down your answer, then select the<strong> solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> Multiplying \(x^2\) by different positive numbers changed the shape of the graph, making it
  "steeper" when the constant is larger and less steep when the constant is smaller. The size of the coefficient
  determined how much of a vertical stretch was applied to the graph.</p>
<ol class="os-raise-noindent" start="3">
  <li>Multiply \(x^2\) by different negative coefficients less than or equal to -1 (for example: \(-x^2\),\(-4x^2\)).
  </li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> Multiplying \(x^2\) by a negative number flipped the graph (or reflected the graph) over the
  \(x\)-axis so that it opens downward rather than upward.</p>
<ol class="os-raise-noindent" start="4">
  <li>Multiply \(x^2\) by different coefficients between -1 and 1 (for example: \(12x^2\), \(-0.25x^2\)).</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>Multiplying \(x^2\) by numbers between -1 and 1 changed the opening of the graph as well as
  the direction of the graph. Shrinking the coefficient from 1 toward 0 made the graph "shallower" and wider. This
  change vertically compressed the graph. When the coefficient is 0, the graph becomes a horizontal line. Raising the
  coefficient from -1 to 0 had the same effect of vertically compressing the graph, except that the graph started out
  opening downward.</p>
<ol class="os-raise-noindent" start="5">
  <li>Add different constants to \(x\) before squaring the term (for example \((x-1)^2\), \((x+2)^2\), \((x-3)^2\)).
  </li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> When a constant is added to the \(x\) in \(x^2\), the graph moves horizontally to the left.
  When a constant is subtracted from the \(x\) in \(x^2\), the graph moves horizontally to the right.</p>
<ol class="os-raise-noindent" start="6">
  <li>Multiply \(x\) by different positive coefficients greater than 1 (for example: (\(3x)^2\), \((7.5x)^2\)).</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> Multiplying \(x\) by different positive numbers changed the shape of the graph, making it
  more narrow when the constant is larger and less narrow when the constant is smaller.</p>
<ol class="os-raise-noindent" start="7">
  <li>Multiply \(x\) by different positive coefficients between 0 and 1 (for example: \((0.5x)^2\), \((0.1x)^2\)).</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> Multiplying \(x\) by different positive numbers between 0 and 1 changed the shape of the
  graph, making it wider as the decimal gets closer to zero.</p>
<p>For quadratics, the parent function is \(f(x)=x^2\) . From there, any moves created by making changes to the function
  like in questions 1&ndash;7 above are called transformations.&nbsp;</p>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>Examine the graphs of three quadratic functions.</p>
<p><img
    alt="3 parabolas on x and y axes. All are symmetric about \(y\)-axis. 2 open up, have positive \(y\)-intercepts, and 1 is wider than the other. Third parabola opens down, is the widest, and has negative \(y\)-intercepts."
    height="254" src="https://k12.openstax.org/contents/raise/resources/48569d296f9bd951abfa5b6505c837c81f060f63"
    width="254"></p>
<ol class="os-raise-noindent">
  <li>What can you say about the coefficient of \(x^2\) in the expression that defines \(f\) (in black at the top)?</li>
</ol>
<p><strong>Answer:</strong> It is positive because the graph of \(f\) opens upward.</p>
<ol class="os-raise-noindent" start="2">
  <li>What can you say about the coefficient of \(x^2\) in the expression that defines \(g\) (in blue in the middle)?
  </li>
</ol>
<p><strong>Answer:</strong> It is positive because the graph of \(g\) opens upward.</p>
<p>Or</p>
<p>The coefficient is less than the coefficient for the graph of \(f\) (but still greater than 0) because \(g\) is
  wider.</p>
<ol class="os-raise-noindent" start="3">
  <li>What can you say about the coefficient of \(x^2\) in the expression that defines \(h\) (in yellow at the bottom)?
  </li>
</ol>
<p><strong>Answer:</strong> It is negative because the graph of \(h\) opens downward.</p>
<p>Or</p>
<p>The absolute value of the coefficient of \(h\) is less than the absolute value of the coefficients for both \(f\) and
  \(g\) because \(h\) is wider.</p>
<ol class="os-raise-noindent" start="4">
  <li>How do each of the functions compare to each other?</li>
</ol>
<p><strong>Answer:</strong> The coefficients cannot be identified exactly without knowing the coordinates of some points
  on the graph. The coefficient of the squared term of \(f\) is greater than that of \(g\) because the graph of \(f\) is
  narrower, indicating that it is growing faster than the graph of \(g\). The absolute value of the coefficient of \(h\)
  is less than that of \(g\) because the graph of \(h\) is wider, indicating that it is not growing as quickly.</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share their observations, and if possible, demonstrate their experiments for all to see. Tell
  students that people often describe the shape when \(a\) is positive as a parabola that "opens upward" and the shape
  when \(a\) is negative as a parabola that "opens downward."</p>
<p>For each change to the expression (for example, adding a constant, or multiplying \(x^2\) by a positive number) and
  the observed change on the graph, solicit students' ideas about why the graph transformed that way. For example, ask:
  "Why do you think subtracting a number from \(x^2\) moves the graph down?"</p>
<p>Discuss questions such as:</p>
<ul class="os-raise-noindent">
  <li> "The points \((1,1)\),\((2,4)\), and \((3,9)\) are three points on the graph representing \(x^2\). When we add 3
    to \(x^2\), how do the \(y\)-values for \(x=1\),\(x=2\), and \(x=3\) change?" (Their \(y\)-values increase by 3:
    \((1,4)\),\((2,7)\),\((3,12)\).) "What about when we subtract 3 from \(x^2\)?" (They decrease by \(3\):
    \((1,-2)\),\((2,1)\),\((3,6)\).) </li>
  <li> "How do the \(y\)-values change when you multiply \(x^2\) by a positive number, say, 3?" (The \(y\) -values for
    \(3x^2\) triple those of \(x^2\). For \(x=1\),\(x=2\), and \(x=3\), the points will be \((1,3)\),\((2,12)\) and
    \((3,27)\).) </li>
  <li> "How do the tripled \(y\)-values affect the graph? (They stretch the graph up vertically, making the graph appear
    narrower.) </li>
</ul>
<p>To help students make stronger connections between the parameters of a quadratic expression and the features of its
  graph, consider the activity &ldquo;Understanding the Behaviors of a Graph in Relation to Its Quadratic
  Expression&rdquo; in this lesson.</p>
<h3>7.12.2: Self Check&nbsp;</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 is the graph of \(f(x)=(x-4)^2\) transformed from the graph of \(f(x)=x^2\)?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">
        Answers
      </th>
      <th scope="col">
        Feedback
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        Up 4
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This would be \(x^2+4\). Subtracting a constant from \(x\)
        moves the graph of \(y = x^2\) that many places right. The answer is right 4.
      </td>
    </tr>
    <tr>
      <td>
        Down 4
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This would be \(x^2-4\). Subtracting a constant from \(x\)
        moves the graph of \(y = x^2\) that many places right. The answer is right 4.&nbsp;
      </td>
    </tr>
    <tr>
      <td>
        Right 4
      </td>
      <td>
        That&rsquo;s correct! Check yourself: When a constant is being subtracted from the \(x\) in the parent function,
        \(f(x)=x^2\), the graph moves horizontally to the right.
      </td>
    </tr>
    <tr>
      <td>
        Left 4
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: This would be \((x+4)^2\). Subtracting a constant from \(x\)
        moves the graph of \(y = x^2\) that many places right. The answer is right 4.&nbsp;
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>7.12.2: 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.&nbsp;</em></p>
<h4>Graphing Quadratic Functions Using Transformations</h4>
<br>
<div class="os-raise-graybox">
  <p><strong>GRAPH A QUADRATIC FUNCTION OF THE FORM </strong>\(f(x)=x^2+k\) <strong>USING A VERTICAL SHIFT </strong></p>
  <hr>
  <p>The graph of \(f(x)=x^2+k\) shifts the graph of \(f(x)=x^2\) vertically \(k\) units. </p>
  <ul>
    <li>If \(k &gt; 0\), shift the parabola vertically up \(k\) units. </li>
    <li>If \(k &lt; 0\), shift the parabola vertically down \(\vert k\vert\) units.</li>
  </ul>
</div>
<br>
<p><img
  height="254" src="https://k12.openstax.org/contents/raise/resources/a2fcaa4ed02f164aaf13be28bd7fb200cfa8d2e0"
  width="254"></p>
  <br>
  <p>For the functions graphed above, notice that the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>
  
  <ol>
      <li>
          <p>The parabola that has been shifted <strong>vertically up</strong> 3 units represents the function \( f(x) = x^2 + 3 \) since \( k = 3 \) and \( 3 \gt 0 \) (orange, dashed graph).</p>
      </li>
      <li>
          <p>The parabola that has been shifted <strong>vertically down</strong> 2 units represents the function \( f(x) = x^2 - 2 \) since \( k = -2 \) and \( -2 \lt 0 \) (green, dotted graph).</p>
      </li>
  </ol>
  <br>
<div class="os-raise-graybox">
  <p><strong>GRAPH A QUADRATIC FUNCTION OF THE FORM</strong> \(f(x)=(x&minus;h)^2\) <strong>USING A HORIZONTAL SHIFT
    </strong></p>
  <hr>
  <p>The graph of \(f(x)=(x&minus;h)^2\) shifts the graph of \(f(x)=x^2\) horizontally \(h\) units. </p>
  <ul>
    <li>If \(h &gt; 0\), shift the parabola horizontally right \(h\) units. </li>
    <li>If \(h &lt; 0\), shift the parabola horizontally left \(\vert h\vert\) units.</li>
  </ul>
</div>
<br>
<p><img
  height="254" src="https://k12.openstax.org/contents/raise/resources/91e909c941eca894057108c85b7e2fd7b8e753a4"
  width="254"></p>
  <br>
  <p>Again, for the functions graphed above, the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>
  
  <ol>
      <li>
          <p>The parabola that has been shifted <strong>horizontally right</strong> 3 units represents the function \( f(x) = (x - 3)^2 \) since \( h = 3 \) and \( 3 \gt 0 \) (orange, dashed graph).</p>
      </li>
      <li>
          <p>The parabola that has been shifted <strong>horizontally left</strong> 2 units represents the function \( f(x) = (x + 2)^2 \) since \( h = -2 \) and \( -2 \lt 0 \) (green, dotted graph).</p>
      </li>
  </ol>
  <br>
<div class="os-raise-graybox">
  <p><strong>GRAPH OF A QUADRATIC FUNCTION OF THE FORM</strong> \(f(x)=ax^2\) </p>
  <hr>
  <p>The coefficient a in the function \(f(x)=ax^2\) affects the graph of \(f(x)=x^2\) by stretching or compressing it.
  </p>
  <ul>
    <li>If \(0&lt;|a|&lt;1\), the graph of \(f(x)=ax^2\) will be &ldquo;wider&rdquo; than the graph of \(f(x)=x^2\).
    </li>
    <li>If \(|a|&gt;1\), the graph of \(f(x)=ax^2\) will be &ldquo;skinnier&rdquo; than the graph of \(f(x)=x^2\).</li>
  </ul>
</div>
<br>
<p><img
  height="254" src="https://k12.openstax.org/contents/raise/resources/a360a4a00b2e68fe8a1665740e9ca15dbf8c9b18"
  width="254"></p>
<br>

<p>Again, the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>

<ol>
    <li>
        <p>The parabola that has been <strong>vertically compressed</strong> by a scale factor of 2 represents the function \( f(x) = \frac{1}{2}x^2 \) since \( a = \frac{1}{2} \) and \( 0 \lt \frac{1}{2} \lt 1 \) (green, dotted graph).</p>
    </li>
    <li>
        <p>The parabola that has been <strong>vertically stretched</strong> by a scale factor of 3 represents the function \( f(x) = 3x^2 \) since \( a = 3 \) and \( 3 \gt 1 \) (orange, dashed graph).</p>
    </li>
</ol>
<br>
<div class="os-raise-graybox">
  <p><strong>GRAPH OF A QUADRATIC FUNCTION OF THE FORM</strong> \( f(x) = (bx)^2 \)</p>

  <p>The coefficient \( b \) in the function \( f(x) = (bx)^2 \) affects the graph of \( f(x) = x^2 \) by stretching or compressing it.</p>
  
  <ol>
      <li>
          <p>If \( 0 \lt |b| \lt 1 \), the graph of \( f(x) = (bx)^2 \) will appear “wider” than the graph of \( f(x) = x^2 \).</p>
      </li>
      <li>
          <p>If \( |b| \gt 1 \), the graph of \( f(x) = (bx)^2 \) will appear “skinnier” than the graph of \( f(x) = x^2 \).</p>
      </li>
  </ol>
  
</div>
<br>
<p><img
  height="254" src="https://k12.openstax.org/contents/raise/resources/434aa81867e0772fd826a5fd3c5cbc67a03ced6c"
  width="254"></p>
<br>
<p>Again, the parent function, \( f(x) = x^2 \), is graphed using a blue, solid line.</p>

<ol>
    <li>
        <p>The parabola that has been <strong>horizontally compressed</strong> by a scale factor of 3 represents the function \( f(x) = (3x)^2 \) since \( b = 3 \) and \( 3 \gt 1 \) (orange, dashed graph).</p>
    </li>
    <li>
        <p>The parabola that has been <strong>horizontally stretched</strong> by a scale factor of 2 represents the function \( f(x) = \left(\frac{1}{2}x\right)^2 \) since \( b = \frac{1}{2} \) and \( 0 \lt \frac{1}{2} \lt 1 \) (green, dotted graph).</p>
    </li>
</ol>
<br>

<h4>Try It: Graphing Quadratic Functions Using Transformations</h4>
<p>How does the graph of \(f(x)=x^2+3\) change from \(f(x)=x^2\)?</p>
<p>Write down your answer, then select the<strong> solution </strong>button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to determine shifts in graphs:</p>
<p>The function \(f(x)=x^2\) is the parent function for quadratics. For \(f(x)=x^2+3\) , when 3 is added to
  \(f(x)=x^2\), the graph shifts up 3 units. This shift occurs because the value of 3 is being added to the output of
  the \(x^2\) function. Thus, the shift is seen vertically because it affects the output values.</p>