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<h4>Activity (15 minutes)</h4>
<p>In this activity, students apply what they learned in earlier lessons to identify the \(x\)-intercepts and the vertex of the graphs representing several quadratic functions. For example, they saw that the \(x\)-coordinate of the vertex is always halfway between those of the \(x\)-intercepts, and the coordinate pairs on one side of the vertex mirror those on the other side. These observations help students locate the vertex of a graph once the \(x\)-intercepts are known, and ultimately to sketch a graph of a quadratic function using at least three identifiable points.</p>
<p>The given expressions here are in factored form, but some are unlike what students have previously seen, so students will need to transfer and generalize the reasoning strategies from earlier work. They also use graphing technology to graph the functions and check their predictions.</p>
<p>In a later lesson, students will use the symmetry of the graph of a quadratic function to sketch graphs when they know the vertex and one additional point rather than the vertex and \(x\)-intercepts.</p>
<h4>Launch</h4>
<p>Arrange students in groups of two to four. Provide access to devices that can run Desmos or other graphing technology. Give students a few minutes of quiet time to think about the first question. Then, ask them to discuss their response and to complete the second question with their group. (Emphasize that students are expected to make the predictions in the first question before using their graphing tool.) To save time, consider asking groups to split the graphing work (each group member graphs only one function and analyze the graphs together).</p>
<p>If students have not yet learned how to use the graphing tools at their disposal to trace a graph or otherwise identify its intercepts and vertex (or other points on the graph), consider demonstrating this during the graphing portion of the activity.</p>
<p>Ask students to attempt the last question (sketching the graph of a function) without using technology. They could use technology to check their sketch afterward.</p>
<br>
<!--BEGIN ELL AND SWD GRAY BOX -->
<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 Collect and Display: Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p> Listen for and collect the language and gestures students use to justify their predictions during small-group discussions about functions \(f\), \(g\), and \(h\). Capture and display language that reflects a variety of ways to determine the coordinates of the points that help them to draw the graph. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.</p>
<p class="os-raise-text-italicize">Design Principle(s): Optimize output (for explanation); Maximize meta-awareness</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>
<!--END ELL AND SWD GRAY BOX -->
<br>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
<li>The functions \(f\), \(g\), and \(h\) are given. Refer to these functions to complete questions a–f.</li>
</ol>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">Equation</th>
<th scope="col">\(x\)-intercept</th>
<th scope="col">\(x\)-coordinate of the vertex</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(f(x)=(x+3)(x-5)\)</p>
</td>
<td>
<ol class="os-raise-noindent" type="a">
<li> _____ </li>
</ol>
</td>
<td>
<ol class="os-raise-noindent" start="2" type="a">
<li> _____ </li>
</ol>
</td>
</tr>
<tr>
<td>
<p>\(g(x)=2x(x-3)\)</p>
</td>
<td>
<ol class="os-raise-noindent" start="3" type="a">
<li> _____ </li>
</ol>
</td>
<td>
<ol class="os-raise-noindent" start="4" type="a">
<li> _____ </li>
</ol>
</td>
</tr>
<tr>
<td>
<p>\(h(x)=(x+4)(4-x)\)</p>
</td>
<td>
<ol class="os-raise-noindent" start="5" type="a">
<li> _____ </li>
</ol>
</td>
<td>
<ol class="os-raise-noindent" start="6" type="a">
<li> _____ </li>
</ol>
</td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Predict the \(x\)-intercepts for function \(f(x)=(x+3)(x-5)\). <br>
<br>
<strong>Answer: </strong>\(x\)-intercepts for function \(f\): \((-3,0)\) and \((5,0)\)
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Predict the \(x\)-coordinate of the vertex for function \(f(x)=(x+3)(x-5)\). <br>
<br>
<strong>Answer:</strong> 1
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> Predict the \(x\)-intercepts for function \(g(x)=2x(x-3)\). <br>
<br>
<strong>Answer: </strong>\(x\)-intercepts for function \(g\): \((0,0)\) and \((3,0)\)
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> Predict the \(x\)-coordinate of the vertex for function \(g(x)=2x(x-3)\). <br>
<br>
<strong>Answer:</strong> 1.5
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li> Predict the \(x\)-intercepts for function \(h(x)=(x+4)(4-x)\). <br>
<br>
<strong>Answer: </strong>\(x\)-intercepts for function \(h\): \((-4,0)\) and \((4,0)\)
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="6" type="a">
<li> Predict the \(x\)-coordinate of the vertex for function \(h(x)=(x+4)(4-x)\). <br>
<br>
<strong>Answer: </strong>0
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="2">
<li>Use the graphing tool or technology outside the course. Graph the functions \(f\), \(g\), and \(h\): <br>
<br>
\(f(x)=(x+3)(x-5)\)<br>
\(g(x)=2x(x-3)\)<br>
\(h(x)=(x+4)(4-x)\)
</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p><strong>Answer:</strong></p>
<p><img alt="3 graphs f, g, h shown on the same coordinate plane." height="219" src="https://k12.openstax.org/contents/raise/resources/1ec4890fea8f856583507ed14d6883ccbf51b444" width="418"></p>
<ol class="os-raise-noindent" start="3">
<li>Without using technology, sketch a graph that represents the equation \(y=(x-7)(x+11)\) and that shows the \(x\)-intercepts and the vertex. Think about how to find the \(y\)-coordinate of the vertex. Be prepared to explain your reasoning.</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p><strong>Answer:</strong> The \(x\)-intercepts for \((x-7)(x+11)\) are at \(x=7\) and \(x=-11\). The vertex for \((x-7)(x+11)\) is halfway between the \(x\)-intercepts, at \(x=-2\). If the \(x\)-coordinate of the vertex is -2, then the \(y\)-value is \((-2,-7)(-2+11)\) or -81. The vertex is at \((-2,81)\). The \(y\)-value when \(x=0\) is \((0-7)(0+11)\), which is -77. The \(y\)-intercept is \((0,-77)\). </p>
<p><img alt="Graph on coordinate plane." height="214" src="https://k12.openstax.org/contents/raise/resources/447d9bc841ac86714834d8cb279b83d898b191ac" width="272"></p>
<h4>Video: Sketching a Graph of a Quadratic Function </h4>
<p>Watch the following video to learn more about sketching a graph of a quadratic function using at least three identifiable points.</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/e54db5db94f260e4358f839163d748806128a386">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/b648d292c43bdf07b745271de3314a418e2139be" srclang="en_us">https://k12.openstax.org/contents/raise/resources/e54db5db94f260e4358f839163d748806128a386
</video></div>
</div>
<br>
<br>
<h4>Student Facing Extension</h4>
<h4>Are you ready for more?</h4>
<p>The quadratic function \(f\) is given by \(f(x)=x^2+2x+6\).</p>
<ol class="os-raise-noindent">
<li>Find \(f(-2)\).<br>
<br>
<strong>Answer:</strong> 6
</li>
</ol>
<ol class="os-raise-noindent" start="2">
<li>Find \(f(0)\).<br>
<br>
<strong>Answer</strong>: 6
</li>
</ol>
<ol class="os-raise-noindent" start="3">
<li>What is the \(x\)-coordinate of the vertex of the graph of this quadratic function?<br>
<br>
<strong>Answer</strong>: -1
</li>
</ol>
<ol class="os-raise-noindent" start="4">
<li>Does the graph have any \(x\)-intercepts? Explain or show how you know.<br>
<br>
<strong>Answer: </strong>No, the graph does not have any \(x\)-intercepts. The \(y\)-coordinate of the vertex is 5 because \(f(-1)=(-1)2+2(-1)+6=5\). The coefficient of the squared term is positive, so the vertex is a minimum, which means the output of the function is never less than 5.
</li>
</ol>
<h4>Activity Synthesis</h4>
<p>Focus the discussion on how students determined the \(x\)-intercepts and the \(x\)-coordinate of the vertex of a graph, and how the coordinates of these points could help them sketch the graph. </p>
<p>Ask questions such as:</p>
<ul>
<li> "For \(g(x)=2x(x-3)\), how did you find the \(x\)-intercepts without graphing?" (The \((x-3)\) suggests that one \(x\)-intercept is \((3,0)\), and evaluating \(g(3)\) does give an output of 0. The factor \(2x\) suggests that the second \(x\)-intercept is \((0,0)\), because \(2(0)\) is 0, and multiplying 0 by any number gives 0.) </li>
<li> "How did you find the \(x\)-value of the vertex?" (By finding the halfway point between 0 and 3, which is 1.5.) </li>
<li> "How would you find the \(y\)-coordinate of the vertex?" (By evaluating \(g(1.5)\), which gives -4.5.) </li>
<li> "The expression that defines function \(h\) has the factor \((4-x)\), where the constant term appears first and \(x\) is subtracted from it. Did this affect how you determined the \(x\)-intercepts? How so?" </li>
<li> "How did you sketch the graph representing \(y=(x-7)(x+11)\)?" (By finding the \(x\)-intercepts and the vertex. The intercepts are \((7,0)\) and \((-11,0)\), so the \(x\)-coordinate of the vertex is -2, and the \(y\)-coordinate is \((-2-7)(-2+11)\) or -81. Those points are enough to sketch a graph.) </li>
<li> (For the Student Facing Extension) “When graphing \(f(x)=x ^2+2x+6\), how does that fact that both \(f(-2)\) and \(f(0)=6\) help you find the \(x\)-coordinate of the vertex?” (Because the graph of a quadratic function is symmetric, any two points that have the same \(y\) value are the same distance from the vertex. You can use this just like you used the halfway point between the zeroes of the function to determine the \(x\)-value of the vertex.) </li>
</ul>
<br>
<h3>7.11.3: 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>
<!--SELF CHECK QUESTION GOES BEFORE THE Table -->
<p class="os-raise-text-bold">QUESTION:</p>
<p>Find the coordinate of the vertex of \(f(x) = (x - 2)(x - 4)\).</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>\((2, 0)\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This is a zero. The answer is \((3, -1)\).</p>
</td>
</tr>
<tr>
<td>
<p>\((3, -1)\)</p>
</td>
<td>
<p>That’s correct! Check yourself: The vertex is located halfway between the zeros of \(x = 2\) and \(x = 4\). Substitute \(x = 3\) into the function and the result is \(y = -1\).</p>
</td>
</tr>
<tr>
<td>
<p>\((4, 0)\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This is a zero. The answer is \((3, -1)\). </p>
</td>
</tr>
<tr>
<td>
<p>\((0, 8)\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This is the \(y\)-intercept. The answer is \((3, -1)\). </p>
</td>
</tr>
</tbody>
</table>
<br>
<!--END SELF CHECK INTRO BEFORE Tables -->
<br>
<h3>7.11.3: 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>Graphing Quadratics With Points</h4>
<p>The function \(f\) given by \(f(x)=(x+1)(x-1)\) is written in factored form. Recall that this form is helpful for finding the zeros of the function (where the function has the value 0) and telling us the \(x\)-intercepts on the graph representing the function.</p>
<p>Here is a graph representing \(f\). It shows two \(x\)-intercepts at \(x=-1\) and \(x=3\).</p>
<p><img alt=" Graph on coordinate plane." height="263" src="https://k12.openstax.org/contents/raise/resources/447d9bc841ac86714834d8cb279b83d898b191ac" width="317"></p>
<p>If we use –1 and 3 as outputs to \(f\), what are the outputs?</p>
<ul>
<li>\(
f(-1)=(-1+1)(-1-3)=(0)(-4)=0\) </li>
<li> \(f(3)=(3+1)(3-3)=(4)(0)=0\) </li>
</ul>
<p>Because the inputs –1 and 3 produce an output of 0, they are the zeros of the function \(f\). And because both \(x\) values have 0 for their \(y\) value, they also give us the \(x\)-intercepts of the graph (the points where the graph crosses the \(x\)-axis, which always have a \(y\)-coordinate of 0). So, the zeros of a function have the same values as the \(x\)-coordinates of the \(x\)-intercepts of the graph of the function.</p>
<p>The factored form can also help us identify the vertex of the graph, which is the point where the function reaches its minimum value. Notice that the \(x\)-coordinate of the vertex is 1, and that 1 is halfway between –1 and 3. Once we know the \(x\)-coordinate of the vertex, we can find the \(y\)-coordinate by evaluating the function \(f(1)=(1+1)(1-3)=(2)(-2)=-4\). So the vertex is at \((1, –4)\).</p>
<p>When a quadratic function is in standard form, the \(y\)-intercept is clear: its \(y\)-coordinate is the constant term \(c\) in \(ax^2+bx+c\). To find the \(y\)-intercept from factored form, we can evaluate the function at \(x=0\), because the \(y\)-intercept is the point where the graph has an input value of 0: </p>
<p>\(f(0)=(0+1)(0-3)=(1)(-3)=-3\)</p>
<h4>Try It: Graphing Quadratics with Points</h4>
<p>Find the zeros and vertex of the function \(f\) given by \(f(x)=(x-4)(x+2)\). Use the points to graph \(f\).</p>
<p><strong>Answer: </strong></p>
<p>Here is how to find the function’s zeros and vertex:</p>
<p><strong>Step 1</strong> - Set each factor equal to 0 and solve. </p>
<ul>
<li>\(
x - 4 = 0, x = 4\) </li>
<li> \(x + 2 = 0, x = -2\) </li>
</ul>
<p><strong>Step 2</strong> - Write the zeros as points. <br>
\((4, 0) (-2, 0)\) </p>
<p><strong>Step 3</strong> - Find the \(x\)-value halfway between the zeros. <br>
\(x=1\) </p>
<p><strong>Step 4</strong> - Substitute \(x = -1\) into the function to find the vertex. <br>
\(f(1) = (1 - 4)(1 + 2) = (-3)(3) = -9\) <br>
Vertex: \((1,-9)\) </p>
<p><strong>Step 5</strong> - Now, use these points to graph \(f\): </p>
<p><img height="252" src="https://k12.openstax.org/contents/raise/resources/c191713d5b7dbc9d098119cda6d531aa53db5096" width="250"></p>