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<h4>Activity (15 minutes)</h4>
<p>The goal of this activity is to uncover the connections between the \(x\)- and \(y\)-intercepts on the graph and the parameters of quadratic expressions in standard and factored form. Earlier in the unit, students learned that the zeros of a quadratic function are \(x\)-values that produce \(y\)-value of 0, and that the zeros of a function are the \(x\)-coordinates of the \(x\)-intercepts of the graph. This idea comes into focus in this activity.</p>
<p>Note that in the examples given here, the \(y\)-coordinate of the \(y\)-intercept is always equal to the product of the zeros, but this is not the case with all graphs representing quadratic functions. (For example, graph representing \(y=2x^2-8\) intercepts the \(x\)-axis at 2 and – 2, but the \(y\)-intercept is \((0,-8)\).) If students make this observation, consider acknowledging that this seems to be true for these graphs and prompting students to revisit this observation in upcoming lessons (in which students will also study graphs) to see if it is always true.</p>
<h4>Launch</h4>
<p>Arrange students in groups of two. Give students a few minutes of quiet time to complete the first two questions. Then, ask them to discuss their observations with a partner before completing the last question.</p>
<h4>Student Activity</h4>
<p>Here are pairs of expressions in standard and factored forms. Each pair of expressions define the same quadratic function, which can be represented with the given graph.</p>
<p>Identify the \(x\)-intercepts and the \(y\)-intercept of each graph.</p>
<ol class="os-raise-noindent">
<li>Function \(f\)<br>
\(x^2+4x+3\)<br>
\((x+3)(x+1)\)<br>
<br>
<img height="214" src="https://k12.openstax.org/contents/raise/resources/e2c53cebfa2afdfbaca4a503d4d37a53d6b9ad7b" width="268">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Identify the \(x\)-intercepts of the graph. <br>
<br>
<strong>Answer: </strong>Graph of \(f\): \(x\)-intercepts: \((-3,0)\) and \((-1,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph. <br>
<br>
<strong>Answer: </strong>3, Graph of \(f\): \(y\)-intercept \((0,3)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="2">
<li>Function \(g\)<br>
\(x^2-5x+4\)<br>
\((x-4)(x-1)\)<br>
<br>
<img alt="Graph of nonlinear function." height="214" src="https://k12.openstax.org/contents/raise/resources/2ad8325341220851c5e571931fb3d42bbf51f7ed" width="268">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Identify the \(x\)-intercepts of the graph. <br>
<br>
<strong>Answer:</strong> Graph of \(g\): \(x\)-intercepts: \((1,0)\) and \((4,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph. <br>
<br>
<strong>Answer:</strong> 4, Graph of \(g\): \(y\)-intercept \((0,4)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="3">
<li>Function \(h\)<br>
\(x^2-9\)<br>
\((x-3)(x+3)\)<br>
<br>
<img alt="Graph of nonlinear function." height="243" src="https://k12.openstax.org/contents/raise/resources/3749c341186ec3703c5f91135a448865fb22f840" width="268">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Identify the \(x\)-intercepts of the graph. <br>
<br>
<strong>Answer: </strong>Graph of \(h\): \(x\)-intercepts: \((3,0)\) and \((-3,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph. <br>
<br>
<strong>Answer: </strong> – 9, Graph of \(h\): \(y\)-intercept \((0,-9)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="4">
<li>Function \(j\)<br>
\(x^2-5x\)<br>
\( x(x-5)\)<br>
<br>
<img alt="Graph of non linear function." height="219" src="https://k12.openstax.org/contents/raise/resources/3370ac3b3a10422245da4106ef6007e1f3435889" width="268">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Identify the \(x\)-intercepts of the graph. <br>
<br>
<strong>Answer:</strong> Graph of \(j\): \(x\)-intercepts: \((0,0)\) and \((5,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph. <br>
<br>
<strong>Answer:</strong> 0, Graph of \(j\): \(y\)-intercept \((0,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="5">
<li>Function \(k\)<br>
\(5x-x^2\)<br>
\(x(5-x)\)<br>
<br>
<img alt="Graph of nonlinear function." height="219" src="https://k12.openstax.org/contents/raise/resources/0dca8066bb0c4fdfd64558eec56cdfc1f982e436" width="268">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Identify the \(x\)-intercepts of the graph. <br>
<br>
<strong>Answer: </strong>Graph of \(k\): \(x\)-intercepts: \((0,0)\) and \((5,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph. <br>
<br>
<strong>Answer:</strong> 0, Graph of \(k\): \(y\)-intercept \((0,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="6">
<li>Function \(l\)<br>
\(x^2+4x+4\)<br>
\((x+2)(x+2)\)<br>
<br>
<img alt="Graph of non linear function." height="219" src="https://k12.openstax.org/contents/raise/resources/2194c7cf7d0bd7abef8ebee3d10e8d4b7fb536f0" width="268">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Identify the \(x\)-intercept of the graph. <br>
<br>
<strong>Answer: </strong> – 2, Graph of \(l\): \(x\)-intercept \((-2,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph. <br>
<br>
<strong>Answer:</strong> 4, Graph of \(l\): \(y\)-intercept \((0,4)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="7">
<li>What do you notice about the \(x\)-intercepts, the \(y\)-intercept, and the numbers in the expressions defining each function? Make a couple of observations.</li>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong></p>
<ul>
<li> The \(y\)-coordinate of the \(y\)-intercept is the constant term when the function is expressed in standard form. The \(y\)-intercept is \((0,-9)\) for the function \(h\), \(x^2-9\). </li>
<li> The \(x\)-intercepts in each graph seem to match the numbers in the factors of the expressions in factored form, but the signs seem to be the opposite. For example, when the expression is \((x+3)(x+1)\), the \(x\)-intercepts are \((-3,0)\) and \((-1,0)\). </li>
<li> When one of the factors is just a variable (rather than a sum or a difference), one of the \(x\)-intercepts is \((0,0)\). For the function \(i\), \(x(x-5)\), one of the intercepts is \((0,0)\)). </li>
<li> x(x-5) and x(5-x) have the same \(x\)-intercepts, but one graph opens upward and the other opens downward. </li>
<li> When the two factors are the same, there is only one \(x\)-intercept. For the function \(l\), \((x+2)(x+2)\), – 2 is the \(x\)-intercept. </li>
</ul>
<ol class="os-raise-noindent" start="8">
<li>Here is an expression that models function \(p\), another quadratic function: \((x-9)(x-1)\).</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Predict the \(x\)-intercepts of the graph that represents this function. <br>
<br>
<strong>Answer:</strong> The \(x\)-intercepts will be \((9,0)\) and \((1,0)\).
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Identify the \(y\)-value of the \(y\)-intercept of the graph that represents this function. <br>
<br>
<strong>Answer: </strong>9
</li>
</ol>
</ol>
<h4> Video: Determining the X- and \(y\)-intercepts of Quadratic Expressions </h4>
<p>Watch the following video to learn more about the \(x\)- and \(y\)-intercepts of quadratic expressions.</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/30c97217de33b541ddaa8d862086fcf589af8285">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/90b4c63a14f3a3468e5792d8973474a3d8dbe0f4" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/30c97217de33b541ddaa8d862086fcf589af8285
</video></div>
</div>
<br>
<br>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>Find the values of \(a\), \(p\), and \(q\) that will make \(y=a(x-p)(x-q)\) be the equation represented by the graph.</p>
<p><img alt="Parabola. Opens up. Vertex = 2 comma -2. X intercepts = 1 and 3. Y intercept = 6." height="219" src="https://k12.openstax.org/contents/raise/resources/3cbe38797c434d41e67e41b6d110fcb2a4e8236f" width="268"></p>
<ol class="os-raise-noindent">
<li>What is the value of \(a\)?<br>
<br>
<strong>Answer: </strong>\(a=2\)
</li>
</ol>
<ol class="os-raise-noindent" start="2">
<li>What is the value of \(p\)?<br>
<br>
<strong>Answer:</strong> \(p=1\) or \(p=3\)
</li>
</ol>
<ol class="os-raise-noindent" start="3">
<li>What is the value of \(q\) ?<br>
<br>
<strong>Answer:</strong> \(q=1\) or \(q=3\)
</li>
</ol>
<h4>Activity Synthesis</h4>
<p>Invite students to share their observations on how the numbers in the quadratic expressions relate to the intercepts of the graphs. Then, ask them to share their predictions for the \(x\)- and \(y\)-intercepts of the graph of function \(p\), defined by \((x-9)(x-1)\). Discuss with students:</p>
<ul>
<li> “How did you find the \(x\)-intercept of the graph of function \(p\) without graphing?” (By looking at the intercepts in other graphs with similar factors. In the examples, expressions of the form \((x-a)(x-b)\) have \((a,0)\) and \((b,0)\) for the \(x\)-intercepts.) </li>
<li> The \(x\)-intercept occurs when \(y\) is zero. So for \((x+3)(x+1)\) to equal zero, one of the factors has to be zero (if \(ab=0\), then \(a=0\) or \(b=0\)). So if \(x+3=0\), then \(x= – 3\). If \(x+1=0\), then \(x=-1\). This prepares students for problems like \((3x+2)(x-1)\), where the zeros will be \(–\frac23\) and 1. </li>
<li> “How did you find the \(y\)-intercept?” (By writing the expression in the standard form and seeing what the constant term is. By evaluating the expression at \(x=0\).) </li>
</ul>
<p>Demonstrate graphing \(p(x)=(x-9)(x-1)\) using the technology available in your classroom. Point to the intercepts. If using Desmos, show students that you can click on the intercepts to reveal the coordinates of the points.</p>
<p>Remind students that earlier in the unit we learned that the \(x\)-intercepts of a graph tell us the zeros of the function, or the input values that produce an output of 0. Highlight that, because an expression in factored form can tell us about the \(x\)-intercept of the graph, this form is also handy for telling us about the zeros of the function that the expression represents.</p>
<p>At this point, students are just noticing that numbers in the expression have something to do with intercepts on the graph. Tell students that, in the next lesson, we will explore why they are related.</p>
<h3>7.10.3: 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>What are the \(x\)-values of the \(x\)-intercept(s) of </p>
<p>\(y=(x-9)(x+2)\)?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">
<p><strong>Answers</strong></p>
</th>
<th scope="col">
<p><strong>Feedback</strong></p>
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(x = 2, x = – 9\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: What value for \(x\) makes the factors zero? The \(x\)-intercepts are the opposite sign of the numbers in the parenthesis. The answer is \(x = – 2, x = 9\).</p>
</td>
</tr>
<tr>
<td>
<p>\(x = – 2, x = 9\)</p>
</td>
<td>
<p>That’s correct! Check yourself: There is a -9 in \((x-9)\) so the \(x\)-intercept is 9. There is a 2 in \((x+2)\) so the \(x\)-intercept is – 2.</p>
</td>
</tr>
<tr>
<td>
<p>\(x = – 2, x = – 9\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: Check your answers by substituting the values into the factors to see if they equal zero. While \(x = – 2\) is a \(x\)-intercept, \(x = – 9\) is not. The answer is \(x = – 2, x = 9\). </p>
</td>
</tr>
<tr>
<td>
<p>\(x = 9, x=2\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: 9 is one of the intercepts. However, <span>the \(x\)-intercepts are the opposite sign of the numbers in the parenthesis. The answer is \(x = – 2, x = 9\)</span></p>
</td>
</tr>
</tbody>
</table>
<br>
<h3>7.10.3: 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>Finding \(x\)-intercepts Without Graphs</h4>
<p>When a quadratic is in factored form, the \(x\)-intercepts of a quadratic equation will be:</p>
<ul>
<li> the opposite sign of the constant in the factor because if \(x+a=0\) , then \(x = – a\). </li>
<li> \(x = 0\) is an option. </li>
<li> If a factor is repeated, there is only one \(x\)-intercept. </li>
</ul>
<p><strong>Example</strong></p>
<p>Find the \(x\)-intercepts of \(y=(x-3)(x+2)\).</p>
<p>The \(x\)-intercepts occur when \(y = 0\). </p>
<p>Set each factor to 0 and solve.</p>
<p>\(x-3=0\;\;\;\;\;\;\;\;\;\;x+2=0\\\;\;\;\;\;x\;=\;3\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\;=\;–\;2\)</p>
<p>Notice that for the factor \((x-3)\) or \((x-3)(x+(-3)\)), the constant is – 3, so the \(x\)-intercept is \(x = +3\) or the point \((3,0)\).</p>
<p>For the factor \((x+2)\), the constant is \(+2\), so the \(x\)-intercept is \(x = – 2\) or the point \((-2,0)\).</p>
<h4>Try It: Finding \(x\)-intercepts Without Graphs</h4>
<p>Find the \(x\)-intercepts of \(y=(x-1)(x+5)\).</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 find the \(x\)-intercepts:</p>
<p>The \(x\)-intercepts occur when \(y = 0\). </p>
<p>Set each factor to 0 and solve.</p>
<p>\(x-1\;=0\;\;\;\;\;\;\;\;x+5=0\\\;\;\;\;\;\;x\;=\;1\;\;\;\;\;\;\;\;\;\;\;\;x\;=\;–\;5\)</p>
<p>Notice that for the factor \((x-3)\) or \((x-1)(x+(-1)\)), the constant is – 1, so the \(x\)-intercept is \(x = +1\) or the point \((1,0)\).</p>
<p>For the factor \((x+5)\), the constant is +5, so the \(x\)-intercept is \(x= – 5\) or the point \((-5,0)\).</p>