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<h4>Activity (10 minutes)</h4>
<p>This activity reinforces what students learned earlier about the connections between the solutions of a quadratic
equation and the zeros of a quadratic function. Previously, students were given equations and asked to graph them to
determine the number of solutions and their values. Here, they are prompted to work the other way around: to write an
equation to represent a quadratic function with only one solution. To do so, students need to make use of the
structure of the factored form and their knowledge of the zero product property.</p>
<h4>Launch</h4>
<p>Give students access to graphing technology. It can be Desmos or other technology outside the course.</p>
<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 7 Compare and Connect: Representing, Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite student pairs to create a visual display of their equation and graph of a quadratic function with only one zero. Allow students time to quietly circulate and analyze at least two other visual displays in the room. Give students quiet think time to consider how the zero is represented in the equation and graph of the quadratic function. Next, ask students to return to their partner and discuss what they noticed. Listen for and amplify observations that connect the zero of the function with the -intercept of the graph. This will help students make connections between the algebraic and graphical representation of quadratic functions.</p>
<p class="os-raise-text-italicize">Design Principle(s): Optimize output; Cultivate conversation</p>
<p class="os-raise-extrasupport-title">Learn more about this routine</p>
<p>
<a href="https://www.youtube.com/watch?v=PF8fRA107OA;&rel=0" target="_blank">View the instructional video</a>
and
<a href="https://k12.openstax.org/contents/raise/resources/94a1159e7b81493c647515711f325771076d99b8" target="_blank">follow along with the materials</a>
to assist you with learning this routine.
</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/0b8a1a4ac3425e84a1d5452b3a5dffa38deb6b13" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<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">Action and Expression: Provide Access for Physical Action</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Support effective and efficient use of tools and assistive technologies. To use graphing technology, some students may benefit from a demonstration or access to step-by-step instructions. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Memory; Attention</p>
</div>
</div>
<br>
<h4>Student Activity </h4>
<p>In previous lessons, we saw that a quadratic equation can have zero, one, or two solutions.</p>
<ol class="os-raise-noindent">
<li>Sketch graphs that represent three quadratic functions: one that has no zeros, one with one zero, and one with two
zeros.</li>
</ol>
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answers:</strong><br>
No zeros:</p>
<p><img height="262" src="https://k12.openstax.org/contents/raise/resources/a655a0de355fddfd470e6ea05905772ddb33755f" width="300"></p>
<p>One zero:</p>
<p><img height="262" src="https://k12.openstax.org/contents/raise/resources/34231239a973fd94fc82fc36965512282698e375" width="300"></p>
<p>Two zeros:</p>
<p><img height="247" src="https://k12.openstax.org/contents/raise/resources/a77acf9fb1fd35c6277cd1175717ddf23bb9765b" width="300"></p>
<ol class="os-raise-noindent" start="2">
<li>Graph the function defined by \(f(x) = x^2− 2x + 1\). Examine the \(x\)-intercepts of the graph.</li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course.</p>
<p>Explain what the \(x\)-intercepts reveal about the function.</p>
<p><strong>Answer:</strong> The graph only intersects the \(x\)-axis at one point: \((1, 0)\). The function only has one
zero.</p>
<ol class="os-raise-noindent" start="3">
<li>Solve \(x^2− 2x + 1 = 0\) by using the factored form and zero product property. What solutions do you get?
</li>
</ol>
<p><strong>Answer:</strong> Rewrite the standard form to \((x − 1)(x − 1) = 0\). The solutions are \(x=1\)
and \(x=1\). Because the two values are the same, we say that this equation has one solution and that \(x = 1\) is a
double root for the equation.</p>
<ol class="os-raise-noindent" start="4">
<li>Try to find another quadratic function that you think will only have one zero.</li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course.</p>
<p>Graph it to check your prediction, then enter your final answer.</p>
<p><strong>Answer:</strong> For example: \(f(x) = (x + 4)(x + 4)\). When graphed, this function only has one zero, at
\(x=-4\).</p>
<br>
<h4>Video: Quadratic Equations with Only One Solution</h4>
<p>Watch the following video to learn more about writing an equation to represent a quadratic function with only one
solution.</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/889b3c40c743e07be9567fff2ea3696f48b5d53f">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/90266f9226968002a7e902339132ae24e25c7831" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/889b3c40c743e07be9567fff2ea3696f48b5d53f
</video></div>
</div>
<br>
<br>
<h4>Activity Synthesis</h4>
<p>Invite students to share how they solved the equation algebraically. Next, invite students to share the equations
they generated. Record and display them for all to see.</p>
<p>Students most likely have written equations in the form of \(g(x) = (x + m)(x + m)\). Ask students why the factored
form, rather than the standard form, might have been preferred. Highlight that by using the same expression for the
two factors, we know that the solution to \((x + m)(x + m) = 0\) will be a single number.</p>
<p>Ask students to describe the graph of a quadratic function with one solution. Point out that this means that the
function will have only one zero, and the graph of the function will have a single horizontal intercept.</p>
<h3>8.9.3: 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>Without graphing, identify the function whose graph has only one zero.</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(f(x) = x^2− 7x − 18\)
</td>
<td>
Incorrect. Let's try again a different way: The factored form of this function is \((x − 9)(x + 2)\).
Since the two solutions are \(x = 9\) and \(x = -2\), the graph of the function has two zeros. The answer is
\(f(x) = x^2−14x + 49\).
</td>
</tr>
<tr>
<td>
\(f(x) = x^2− 1\)
</td>
<td>
Incorrect. Let's try again a different way: The factored form of this function is \((x − 1)(x + 1)\).
Since the two solutions are \(x = 1\) and \(x = -1\), the graph of the function has two zeros. The answer is
\(f(x) = x^2−14x + 49\).
</td>
</tr>
<tr>
<td>
\(f(x) = x^2−14x + 49\)
</td>
<td>
That's correct! Check yourself: The factored form of \(x^2−14x + 49\) is \((x − 7)(x −
7)\). Since there is only one solution, \(x = 7\), the graph of the function only has one zero.
</td>
</tr>
<tr>
<td>
\(f(x) = x^2+ 2x − 15\)
</td>
<td>
Incorrect. Let's try again a different way: The factored form of this function is \((x − 3)(x + 5)\).
Since the two solutions are \(x = 3\) and \(x = -5\), the graph of the function has two zeros. The answer is
\(f(x) = x^2−14x + 49\).
</td>
</tr>
</tbody>
</table>
<br>
<h3>8.9.3: 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>Writing an Equation to Represent a Quadratic Function with Only One Solution</h4>
<p class="os-raise-text-bold">Identifying the Number of Zeros</p>
<p>How many zeros does the graph of the function \(f(x) = x^2+ 5x − 14\) have? How can you find out without
graphing the function?</p>
<p>Let's find the factored form of the function and solve it.</p>
<p><strong>Step 1</strong> - Set the function equal to 0.</p>
<p> \(x^2+ 5x − 14 = 0\) </p>
<p><strong>Step 2</strong> - Find the factored form.</p>
<p> \((x + 7)(x − 2) = 0\) </p>
<p><strong>Step 3</strong> - Set each factor equal to 0 and solve.</p>
<p> \(x = -7\) and \(x = 2\)</p>
<p>Since there are two solutions, we know that the function \(f(x) = x^2+ 5x − 14\) will have zeros at \(x=-7\)
and \(x=2\).</p>
<p>In other words, the \(x\)-intercepts of the function are \((-7, 0)\) and \((2, 0)\).</p>
<p>For any function, the number of solutions is equal to the number of zeros when graphed.</p>
<p class="os-raise-text-bold">Writing an Equation to Represent a Quadratic Function with One Solution</p>
<p>We can also find a quadratic function if we know the zeros. For example, if we want to write a quadratic function
that has one zero or one solution, we can start with a factored form of the function representing one zero or
solution.</p>
<p>Any factored form of a function with a factor that repeats itself, such as \((x − 3)(x − 3)\), has only
one solution or one zero.</p>
<p>We can graph the function \(f(x) = (x − 3)(x − 3)\) to check our prediction.</p>
<p>We know the solution is \(x=3\). Let's rewrite the function in standard form.</p>
<p>Factored form:<br>
\(f(x) = (x − 3)(x − 3)\) </p>
<p>Multiply to find standard form:<br>
\(f(x) = x^2− 6x + 9\) </p>
<p>Now, we graph the function \(f(x) = x^2− 6x + 9\).</p>
<p><img alt="GRAPH OF A PARABOLA THAT OPENS UP WITH A VERTEX AT THE POINT (3, 0)." height="410" src="https://k12.openstax.org/contents/raise/resources/c8b227aec550d08016f6927c0d4cc47ce1c1e89c" width="400"></p>
<p>The graph of the function has one zero at \(x=3\). The \(x\)-intercept of the function is \((3, 0)\). The equation is
said to have a double root at this point.</p>
<p>So, our prediction was correct.</p>
<h4>Try It: Writing an Equation to Represent a Quadratic Function with Only One Solution</h4>
<ol class="os-raise-noindent">
<li>Write a quadratic function in standard form whose graph has two zeros.</li>
</ol>
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to write the quadratic functions in standard form:</p>
<p>The quadratic function has two solutions.</p>
<p>This means the factors in factored form are not equal. We could use:<br>
\(f(x) = (x + 5)(x + 2)\).</p>
<p>Multiply to find standard form.<br>
\(f(x) = x^2+ 7x + 10\) </p>
<p>The graph of \(f(x) = x^2+ 7x + 10\) has two zeros. The \(x\)-intercepts will be \((-5, 0)\) and \((-2, 0)\).</p>
<br>
<ol class="os-raise-noindent" start="2">
<li>Write a quadratic function in standard form whose graph has one zero.</li>
</ol>
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to write the quadratic functions in standard form:</p>
<p>The quadratic function has one solution.</p>
<p>That means the factors in factored form are equal. We could use:<br>
\(f(x) = (x + 10)(x + 10)\).</p>
<p>Multiply to find standard form.<br>
\(f(x) = x^2+ 20x + 100\) </p>
<p>The graph of \(f(x) = x^2+ 20x + 100\) has one zero. The \(x\)-intercept will be \((-10, 0)\).</p>