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<h4>Activity (15 minutes)</h4>
<p>In this activity, students apply what they know about the vertex form to modify a given quadratic expression such that its graph is translated in particular ways to produce a different vertex. Students also consider (by analyzing an argument) whether modifying the squared term of a quadratic expression in standard form produces the same translation on the graph as modifying the squared term of an expression in vertex form.</p>
<p>As they modify expressions to translate graphs, students make use of structure. As they analyze an argument about how changing a quadratic expression in standard form affects the graph, they practice explaining their thinking and critiquing the reasoning of others.</p>
<h4>Launch</h4>
<p>Remind students that they have studied the connections between quadratic expressions and their graphs over several lessons. They have matched expressions and graphs and sketched graphs of given expressions. Now they will think about how to change one or more parts of a quadratic expression so that its graph has certain features.</p>
<p>Before beginning the activity, ask students to describe the graph representing \(y=x^2\). Make sure students recall that the graph has a vertex at the origin and opens up. If needed, ask them to name some points that are on the graph, for example, \((-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9)\).</p>
<p>Give students access to graphing technology, but tell students to answer the first question without using technology.</p>
<p>Consider arranging them in groups of two and encouraging them to discuss the last question with a partner after some quiet time to think individually. If students use a graph to illustrate that Kiran's statement is incorrect, encourage them to also use the expression in their justification.</p>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
<li>How would you change the equation \(y=x^2\) so that the vertex of the graph of the new equation is located at the following coordinates and the graph opens as described?</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>\((0,11)\), opens up </li>
</ol>
<p><strong>Answer:</strong> \(y=x^2+11\)</p>
<ol class="os-raise-noindent" start="2" type="a">
<li> \((7,11)\), opens up </li>
</ol>
<p><strong>Answer:</strong> \(y=(x−7)^2+11\)</p>
<ol class="os-raise-noindent" start="3" type="a">
<li>\((7,-3)\), opens down </li>
</ol>
<p><strong>Answer:</strong> \(y=-1(x−7)^2−3\)</p>
<ol class="os-raise-noindent" start="2">
<li>Use the graphing tool or technology outside the course. Verify your predictions.</li>
</ol>
<p>Select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong></p>
<p><img height="320" src="https://k12.openstax.org/contents/raise/resources/4843161eef643f7108bb7fb8af0693b68859ca05" width="312"></p>
<ol class="os-raise-noindent" start="3">
<li>Kiran graphed the equation \(y=x^2+1\) and noticed that the vertex is at \((0,1)\). He changed the equation to \(y=(x−3)^2+1\) and saw that the graph shifted 3 units to the right and the vertex is now at \((3,1)\).</li>
</ol>
<p>Next, he graphed the equation \(y=x^2+2x+1\) and observed that the vertex is at \((-1,0)\). Kiran thought, "If I change the squared term \(x^2\) to \((x−5)^2\), the graph will move 5 units to the right and the vertex will be at \((4,0)\)."</p>
<ol class="os-raise-noindent" type="a">
<li> Do you agree with Kiran?
<ul>
<li> Agree </li>
<li> Disagree </li>
</ul>
</li>
</ol>
<p><strong>Answer: </strong>Disagree</p>
<ol class="os-raise-noindent" start="2" type="a">
<li> Explain or show your reasoning why you agree or disagree with Kiran. </li>
</ol>
<p><strong>Answer:</strong> You should disagree with Kiran.</p>
<p><img height="312" src="https://k12.openstax.org/contents/raise/resources/8257db4bc20f3131296727fde38411df9fe0e233" width="312"></p>
<ul>
<li> The vertex of the graph of the original equation \(y=x^2+2x+1\) is at \((-1,0)\). The vertex of \(y=(x−5)^2+2x+1\) is at \((4,10)\). </li>
<li> Evaluating \(y=(x−5)^2+2x+1\) at \(x = 4\) gives \(y=10\), not \(y=0\). </li>
<li> The original equation is not in vertex form, so changing \(x^2\) into \((x−5)^2\) does not change the location of the vertex 5 units to the right. </li>
</ul>
<ol class="os-raise-noindent" start="4">
<li>\(y=(x+4)^2-1\) is a parabola with a vertex at \((-4,1)\) that opens upward.</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li> If we multiply \(y\) by 2, we get \(2y=2[(x+4)^2-1] = 2(x+4)^2-2\). How does this change the graph? </li>
</ol>
<p><strong>Answer: </strong>The parabola is narrower and the vertex is lower. The \(x\)-intercepts stay the same.</p>
<ol class="os-raise-noindent" start="2" type="a">
<li> If we multiply \(y\) by \(\frac{1}{2}\), we get \(y=\frac{1}{2}[(x+4)^2-1] =\frac{1}{2}(x+4)^2-\frac{1}{2}.\) How does this change the graph?</li>
</ol>
<p><strong>Answer: </strong>The parabola is wider and the vertex is higher. The \(x\)-intercepts stay the same.</p>
<ol class="os-raise-noindent" start="5">
<li> Verify your prediction using the graphing tool or technology outside the course. <br>
(Students were provided access to Desmos)</li>
</ol>
<p><strong>Answer: </strong></p>
<img alt="Graph of the equations y equals x plus four squared minus one, y equals two times x plus 4 squared minus two, and y equals twelve times x plus 4 squared minus twelve" height="312" src="https://k12.openstax.org/contents/raise/resources/faff41a5e75f54fcfc04529b6122df184030cb64">
<br>
<br>
<ol class="os-raise-noindent" start="6">
<li>Sophia graphed the equation \(y=x^2-6x+9\). She wondered what would happen to the graph if everywhere she had an \(x\), she replaced it with \(3x\). </li>
</ol>
<ol class="os-raise-noindent" type="a">
<li> Sophia predicted that the parabola would be narrower, and the \(x\)-intercepts and \(y\)-intercepts would change. Write why you agree or disagree with Sophia’s prediction.</li>
</ol>
<p><strong>Answer: </strong>The parabola will be narrower and the \(x\)-intercepts will change, but the \(y\)-intercept will stay the same.</p>
<ol class="os-raise-noindent" start="2" type="a">
<li> Sophia also wondered what would happen if she replaced the \(x\) in \(y=x^2-6x+9\) with \(\frac{1}{3}x\), how that would change the graph. She predicted that the parabola would be wider, and the \(x\)-intercepts and \(y\)-intercepts would change. Write why you agree or disagree with Sophia’s prediction.</li>
</ol>
<p><strong>Answer: </strong>The parabola will be wider and the \(x\)-intercepts will change, but the \(y\)-intercept will stay the same.</p>
<ol class="os-raise-noindent" start="7">
<li> Verify your prediction using the graphing tool or technology outside the course. <br>
(Students were provided access to Desmos)</li>
</ol>
<p><strong>Answer: </strong></p>
<img alt="Graph of the equations y equals x squared minus six x plus 9, y equals three x squared minus eighteen x plus 9, and y equals thirteen x squared minus seventy-eight x squared plus 9" height="312" src="https://k12.openstax.org/contents/raise/resources/a5cf564d8460ded5b99a7671f653edecc21a8a49">
<br>
<br>
<h4>Activity Synthesis</h4>
<p>Select students to share their equations for the first question and their explanations for how they knew what modifications to make.</p>
<p>Then, focus the discussion on the third question and how students knew that when a quadratic expression is in standard form, adding a constant term before squaring the input variable does not translate the graph the same way as when the expression is in vertex form. At this point, students are not expected to come up with a rigorous justification as to why the graph will not translate as Kiran described. They are only to make this observation and consider ways to explain it.</p>
<p>Solicit as many explanations as time permits. If no one mentioned that the expression \(x^2+2x+1\) is not in vertex form and that its parameters do not relate to the graph the same way, bring these points up.</p>
<p>If time permits, consider pointing out that in \(x^2+2x+1\), the input \(x\) shows up in the squared term and the linear term. If we subtract 5 from \(x\) before it is squared, but do not subtract 5 from \(x\) before it is multiplied by 2, then the graph does not shift horizontally. (If we graph \((x−5)^2+2(x−5)+1\), the graph does shift 5 units to the right.) In a later course, students will look more closely at the effects on the graph of replacing \(f(x)\) by, for instance, \(f(x)+k\) and \(f(x+k)\).</p>
<h3>7.17.2: Self Check</h3>
<!--BEGIN SELF CHECK INTRO BEFORE Tables -->
<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>What transformations and shifts need to be made to change \(y=x^2\) to \(y=-(x+5)^2-10\)?</p>
<!--SELF CHECK table-->
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>Right 5 and down 10</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: To move the graph to the right, 5 would have to be added to \(x\). Also, notice the coefficient of the squared term is negative. The answer is it reflects over the \(x\)-axis, moves left 5, and down 10.</p>
</td>
</tr>
<tr>
<td>
<p>Left 5 and down 10</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: The equation leads with a negative. The answer is it reflects over the \(x\)-axis, moves left 5, and down 10.</p>
</td>
</tr>
<tr>
<td>
<p>Reflect over the \(x\)-axis, left 5, and down 10</p>
</td>
<td>
<p>That’s correct! Check yourself: The negative in the front causes the parabola to reflect over the \(x\)-axis. The vertex will be at \((-5, -10)\) so the graph of \(y=x^2\) shifts left 5 and down 10.</p>
</td>
</tr>
<tr>
<td>
<p>Reflect over the \(x\)-axis, left 5, and up 10</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: The constant term is negative, so the graph shifts down. The answer is it reflects over the \(x\)-axis, moves left 5, and down 10.</p>
</td>
</tr>
</tbody>
</table>
<br>
<!--END SELF CHECK INTRO BEFORE Tables -->
<br>
<h3>7.17.2: Additional Resources</h3>
<p><strong><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></strong></p>
<h4>Transforming Graphs of Quadratics</h4>
<p>The graphs of \(y=x^2\), \(y=x^2+12\), and \(y=(x+3)^2\) all have the same shape but their locations are different. The graph that represents \(y=x^2\) has its vertex at \((0, 0)\).</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/d876c425fd41cfada69fb6bf00502ec7e3e3c2b5" alt="Three parabolas in x y plane, origin O."/></p>
<p>Notice that adding 12 to \(x^2\) raises the graph by 12 units, so the vertex of that graph is at \((0, 12)\). Replacing \(x^2\) with \((x+3)^2\) shifts the graph 3 units to the left, so the vertex is now at \((-3, 0)\).</p>
<p>We can also shift a graph both horizontally and vertically.</p>
<p>The graph that represents \(y=(x+3)^2 +12\) will look like that for \(y=x^2\) but it will be shifted 12 units up and 3 units to the left. Its vertex is at \((-3, 12)\).</p>
<p><img alt="Two parabolas in x y plane, origin O." src="https://k12.openstax.org/contents/raise/resources/74c7e56b30ffad5602a6859500e4f1f324292a73"></p>
<p><br>
</p>
<div class="os-raise-graybox os-raise-text-center">
<p class="os-raise-text-bold">Translating Functions </p>
<p>A vertical shift transforms the parent function into another function by moving the graph up or down “d” units. </p>
<p>A horizontal shift transforms the parent function into another function by moving the graph left or right “c” units. </p>
</div>
<br>
<p>The graph representing the equation \(y=-(x+3)^2+12\) has the same vertex at \((-3, 12)\), but because the squared term \((x+3)^2\) is multiplied by a negative number, the graph is flipped over horizontally, so that it opens downward.</p>
<p><img alt="Two parabolas in x y plane, origin O." src="https://k12.openstax.org/contents/raise/resources/b87f2134a0b187a0d797ffbd902d5d37ad30055a"></p>
<p>Quadratic equations can also be transformed through dilations that stretch or compress the parabola. The graphs for the parent quadratic function, \(y = x^2)\), and \(y = 4x^2\), and \(y = \frac15x^2\) are all parabolas. The second and third equation, however, have been vertically dilated from the parent function.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/e2ac8948c5a1185af5a216e6a2ba21b40e26d634" width="300"/></p>
<p>The orange dashed parabola represents \(y = 4x^2\) and depicts the result of multiplying each output of the function by 4. This dilation represents a vertical stretch.</p>
<p>The green dotted parabola represents \(y = \frac 15x^2\) and depicts a graph where each of the output values from the function are multiplied by \(\frac15\). This dilation represents a vertical compression.</p>
<p>The scale factor of the coefficient we multiply by determines the impact on the graph. If the scale factor is greater than 1, then the dilation is a vertical stretch. If the scale factor is between 0 and 1, then the dilation is a vertical compression.</p>
<p>When quadratic equations are dilated horizontally, the result looks similar but the impact to the function is very different because horizontal dilations affect the input values before the function is applied. The graphs for the parent quadratic function, \(y =x^2\), and \(y = (4x)^2\) and \(y = (\frac13x)^2\) are displayed below.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/c529e3fa260955be35761a92a99d58362d4e0518" width="300"/></p>
<p>The orange dashed parabola represents \(y = (4x)^2\) and depicts the result of multiplying each input of the function by 4 and then squaring. This dilation represents a horizontal compression.</p>
<p>The green dotted parabola represents \(y = (\frac 13x)^2\) and depicts a graph where each of the input values from the function are multiplied by \(\frac13\). This dilation represents a horizontal stretch.</p>
<p>Notice that the scale factor determines the impact on the graph here, too. If the scale factor is greater than 1, then the dilation is a horizontal compression. If the scale factor is between 0 and 1, then the dilation is a horizontal stretch.</p>
<br>
<div class="os-raise-graybox os-raise-text-center">
<p class="os-raise-text-bold">Dilating Functions </p>
<p>\(a f(bx)\)</p>
<p>A vertical dilation transforms the parent function into another function by stretching or compressing the output values by a scale factor of “\(a\).”</p>
<p>A horizontal dilation transforms the parent function into another function by stretching or compressing the input values by a scale factor of “\(b\).”</p>
</div>
<br>
<br>
<h4>Try It: Transforming Graphs of Quadratics</h4>
<ol class="os-raise-noindent">
<li>What transformations and shifts need to be made to change \(y=x^2\) to \(y=-(x-2)^2+7\)?</li>
</ol>
<h5>Solution</h5>
<p>Here is how to determine the transformations needed:</p>
<p><strong>Step 1 -</strong> Identify the vertex of the new function.<br>
\((2, 7)\)</p>
<p><strong>Step 2 -</strong> Identify the shifts needed to go from the origin \((0,0)\) to the new vertex.<br>
Right 2, up 7</p>
<p><strong>Step 3 -</strong> Ask: Does the parabola open up or down? Does it reflect over the \(x\)-axis?<br>
Opens down since the equation leads with a negative. Reflects over the \(x\)-axis. </p>
<p>The transformations needed are right 2, up 7, and a reflection over the \(x\)-axis.</p>
<br>
<ol class="os-raise-noindent" start="2">
<li>What transformations need to be made to \(y =x^2\) so it becomes \(y = (3x)^2\)?</li>
</ol>
<h5 >Solution</h5>
<p>The needed change impacts the \(x\)-values and follows the rule for a horizontal dilation, \(f(bx)\). This dilation represents a horizontal compression by a scale factor of 3. </p>