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<h4>Activity (15 minutes)</h4>
<p>This activity serves two main goals. It prompts students to look closely at appropriate domains for quadratic functions given the situations they represent. It also gives them an opportunity to identify or estimate the vertex of the graph and the zeros of the functions, and then to interpret them in context.</p>
<h4>Launch</h4>
<p>In the previous activity, we saw a function representing the revenue, in thousands of dollars, from selling new movies at \(x\) dollars. If we graph the equation \(r=x(18-x)\) using graphing technology, it would produce a graph like this:</p>
<p><img alt='Graph with \(x\)-axis labeled with price in dollars and \(y\)-axis labeled with revenue in thousands of dollars." height="201' height="268" src="https://k12.openstax.org/contents/raise/resources/763a11b765e8a6f0000553133d0269bb64ddafa3" width="434"></p>
<p>Then, discuss questions such as:</p>
<ul>
<li> “What domain is appropriate for the price of a new movie?” (The price can’t be negative and it can’t be more than \($18\), or else the revenue will be negative, which is not possible, assuming they are selling a non-negative number of downloads for a non-negative price. So an appropriate domain would be \(18 \leq x \leq 18\). The part of the graph to the left of the vertical axis has no meaning because we can’t have a negative price. The part representing a price above \($18\) would mean the company is losing money, so it has meaning, but it is unlikely to be considered if the company is trying to maximize revenue.) </li>
<li> “What are the zeros of the function? What do they tell us in this situation?” (\(0\) and \(18\). They tell us the prices at which the company would earn no revenue.) </li>
<li> “What is the vertex of the graph representing the function? What does it tell us in this situation?” (It appears to be at \((9,81)\). It tells us the price that would produce the greatest revenue.) </li>
</ul>
<p>Tell students they will now think about the domain, vertex, and zeros of a few quadratic functions we have seen so far.</p>
<p>Arrange students in groups of two to four. Consider asking each group to work on only 1–2 functions and then to share their findings with the class, or choosing only a couple of functions for the class to investigate. If the activity is divided among groups and if time permits, consider asking each group to prepare a presentation or to display their work for a gallery walk.</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. Ask students to select one of the functions and to prepare a visual display of their work. Students should consider how to display their reasoning so that another student can interpret what they see. Suggest that students should add notes and details to the graphs or functions to help communicate their thinking. Arrange students in groups of two, and provide two or three minutes of quiet think time for students to read and interpret each other's work before a whole-class discussion.</p>
<p class="os-raise-text-italicize">Design Principle(s): Cultivate conversation; Maximize meta-awareness</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>
<h4>Student Activity</h4>
<p>Here are three sets of descriptions and equations that represent some familiar quadratic functions. The graphs show what a graphing technology may produce when the equations are graphed. For a more accurate answer, use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool below.
<br>(Students had access to Desmos.)
</p>
<ul>
<li> Describe a <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> domain</span></strong> that is appropriate for the situation. Think about any upper or lower limits for the input, as well as whether all numbers make sense as the input. Write the domain as an inequality. Then, describe how the graph should be modified to show the domain that makes sense. </li>
<li> Describe a <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> range</span></strong> that is appropriate for the situation. Think about any upper or lower limits for the input, as well as whether all numbers make sense as the input. Write the range as an inequality. What does the range mean in this situation? Then, describe how the graph should be modified to show the range that makes sense.</li>
<li> Identify or estimate the The <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> vertex (of a graph)</span></strong>. Describe what it means in the situation. </li>
<li> Identify or estimate the <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> zero of the function</span></strong>. Describe what they mean in the situation. </li>
</ul>
<p>Use the following information to answer questions 1-4.</p>
<blockquote>
<p> The area of a rectangle with a perimeter of \(25\) meters and a side length \(x\): \(A(x)=x \cdot \frac {(25-2x)}{2}\)</p>
<p><img alt='Graph of the quadratic function <span class="math" data-png-file-id="35294"></span> on a coordinate plane, origin <span class="math" data-png-file-id="35"></span>. Horizontal axis scale negative 10 to 15 by 5s, labeled “length (meters)”. Vertical axis scale 0 to 60 by 10’s, labeled “area (square meters)”. Some of the points of this function are <span class="math" data-png-file-id="35"></span>(0 comma 0), (1 comma 11 point 5) and (5 comma 37 point 5) to a maximum at (6 point 2 5 comma 39 point 0 6 3) then decreasing through (10 comma 25), (12 comma 6) and (12 point 5 comma 0).' height="303" src="https://k12.openstax.org/contents/raise/resources/9b71518a180e946db0e0245d8ef9f00e5121fd3e" width="279"></p>
</blockquote>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent">
<li> Domain: </li>
<br>
<p><strong>Answer:</strong> \(0<x<12.5\). All positive rational numbers make sense as an input. The part of the graph that is below the horizontal line is not meaningful here (negative lengths or lengths greater than \(12.5\) produce a negative area, which doesn’t make sense). Having a length or area equal to \(0\) doesn’t make sense either.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2">
<li>What is an inequality that represents the domain? </li>
<br>
<p><strong>Answer:</strong> \(0<y<39.06\). All positive rational numbers make sense as an output up to the maximum of the graph at 39.06. The part of the graph that is below the horizontal line is not meaningful here since there can not be a negative area (which doesn’t make sense). Having a length or area equal to 0 doesn’t make sense either.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3">
<li>What is an inequality that represents the range? </li>
<br>
<p><strong>Answer:</strong> The vertex is the point on the graph when the input is at \(6.25\). It’s around \((6.25,39.063)\). It’s when the rectangle has the greatest area.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4">
<li> What are the zeros of the graph? </li>
<br>
<p><strong>Answer:</strong> The zeros are \(0\) and \(12.5\). They tell us that when the side length of the rectangle is \(0\) or \(12.5\) meters, the rectangle has no area (no rectangle exists).</p>
</ol>
</ol>
<p>Use the following information to answer questions 5-8.</p>
<blockquote>
<p>The number of squares as a function of step number \(n\): \(f(n)=n^2+4\)</p>
<p><img alt='Graph of the quadratic function&nbsp;f of n = n squared + 4.&nbsp;Horizontal axis scale negative 12 to 12 by 6’s, labeled “step number”. Vertical axis scale 0 to 200 by 40’s, labeled “number of squares”.&nbsp;" height="222' height="296" src="https://k12.openstax.org/contents/raise/resources/7b7b6180d83187d6cbb0fc79923a160133ec5e0e" width="262"></p>
</blockquote>
<br>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5">
<li>What is an inequality that represents the domain? </li>
<br>
<p><strong>Answer:</strong> \(n \geq 0\) where \(n\) is a whole number. Step numbers that are negative or decimal don’t make sense here. The graph should show points at non-negative integer values (instead of a continuous curve). Everything to the left of the vertical axis does not have meaning here.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="6">
<li>What is an inequality that represents the range? </li>
<br>
<p><strong>Answer:</strong> \(f≥4\). The number of squares starts with 4 and increases.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="7">
<li> Vertex: </li>
<br>
<p><strong>Answer:</strong> The vertex is \((0,4)\). It tells us the number of squares at Step \(0\).</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="8">
<li> Zeros: </li>
<br>
<p><strong>Answer:</strong> There are no zeros for this function. There is not a time when there are no squares in the pattern.</p>
</ol>
</ol>
<p>Use the following information to answer questions 9-12.</p>
<blockquote>
<p>The distance in feet that an object has fallen \(t\) seconds after being dropped: \(g(t)=16t^2\)</p>
<p><img alt=' <p>Graph of the quadratic function <span class="math" data-png-file-id="35295"></span> on a coordinate plane, origin <span class="math" data-png-file-id="35"></span>. Horizontal axis scale negative 8 to 8 by 4’s, labeled “time (seconds)”. Vertical axis scale 0 to 600 by 200’s, labeled “distance fallen (feet)”. Some of the points of this function are (negative 8 comma 1024), (negative 5 comma 400) and (negative 1 comma 16) to a minimum at <span class="math" data-png-file-id="35"></span>(0 comma 0) then increasing through (1 comma 16), (5 comma 400) and (8 comma 1024).</p>' height="287" src="https://k12.openstax.org/contents/raise/resources/874ea7b6b1f39ebb7f8214a821393c0eca2560aa" width="262"></p>
</blockquote>
<br>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="9">
<li>What is an inequality that represents the domain? </li>
<br>
<p><strong>Answer:</strong> \(t \geq 0\) or only positive numbers and \(0\). Negative values of time don’t make sense here, so the part of the graph to the left side of the vertical axis has no meaning.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="10">
<li>What is an inequality that represents the range? </li>
<br>
<p><strong>Answer:</strong> \(g≥0\). The distance fallen increases as time increases and would start at \(g=0\).</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="11">
<li> Vertex: </li>
<br>
<p><strong>Answer:</strong> The vertex is \((0,0)\). It tells us the distance fallen at \(0\) seconds, which is \(0\) feet.</p>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="12">
<li> Zeros: </li>
<br>
<p><strong>Answer:</strong> The vertex also tells us the zero of the function. It’s the only time when the distance fallen is \(0\) feet.</p>
</ol>
</ol>
<h4>Video: Identifying Domain, Vertex, and Zero of Quadratic Functions</h4>
<p>Watch the following video to learn more about the domain, vertex, and zero of quadratic functions.</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/ce57cc6be915599b86183af5e016926306d73f79">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/54ddb28fcbc569717c38850db6b937c7c3bdb75e" srclang="en_us">https://k12.openstax.org/contents/raise/resources/ce57cc6be915599b86183af5e016926306d73f79
</video></div>
</div>
<br>
<br>
<h4>Anticipated Misconceptions</h4>
<p>Some students may confuse zeros and horizontal intercepts (\(x\)-intercepts). Watch for students who write the zeros as ordered pairs such as \((25,0)\) rather than \(25\). Emphasize that while these two terms are related, there is a difference. A zero is an input value that makes the function’s output \(0\), and the horizontal intercept is the point where the graph of the function meets the horizontal axis. A zero of a function is the \(x\)-coordinate of an \(x\)-intercept of its graph.</p>
<h4>Activity Synthesis</h4>
<p>Invite groups to share their responses and explanations. If not already discussed or displayed by students, show examples of graphs that are each adjusted for a domain appropriate for the function represented.</p>
<p>Explain that the graph of a quadratic function may or may not show the vertex, depending on the situation it represents.</p>
<p>Here are graphs representing the functions defined by \(f(n)=n^2+4\) and \(h(t)=576-16t^2\) (in the second and fourth questions), each adjusted for the domain appropriate in the situation. In each graph, the \(y\)-intercept is the vertex, but because a negative domain is not applicable, we don’t see a “turn” in the graph (where the output changes from increasing to decreasing, or vice versa).</p>
<p><img alt='Points on coordinate plane. X-axis labeled with step number and \(y\)-axis labeled with number of squares." height="222' height="296" src="https://k12.openstax.org/contents/raise/resources/61647dba44d7d70758c76917c326cd0f7e636982" width="262"></p>
<br>
<p><img alt="Graph of line. X-axis labeled with time in seconds and \(y\)-axis labeled with height in feet." height="287" src="https://k12.openstax.org/contents/raise/resources/3f02d4c9fe240b174859c9af710600ad4a4d1e89" width="262"></p>
<ul>
<li> The function \(f\) represents the number of squares at step \(n\) of a geometric pattern. Because partial step numbers are not possible, it makes sense for the graph to show discrete points at whole-number input values, rather than a continuous curve that includes all rational numbers for the input. </li>
<li> The function \(h\) models the height of an object \(t\) seconds after being dropped. Because a negative number of seconds is not meaningful here, assuming the object stops once it hits the ground (at 6 seconds), an appropriate domain for the function would be \(0 \leq t \leq 6\). </li>
</ul>
<h3>7.7.3: 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>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Find the approximate domain of the graph below:</p>
<p><img alt="Graph of the quadratic function. h(t)=5+60t−16t^2 on a coordinate plane, origin O. Horizontal axis scale 0 to 4 by 1’s, labeled “time (seconds)”. Vertical axis scale 0 to 80 by 20’s, labeled “distance above ground (feet)”. Some of the points of this function are (0 comma 5), (1 comma 49), to a maximum near (1 point 9 comma 61 point 2 5) then decreasing through (2 comma 61), (3 comma 41) and (3.8 comma 0).
" height="231" src="https://k12.openstax.org/contents/raise/resources/32a38b72c84f4afe7093510818605bcc7a9b6d4f" width="311"></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>\(0 \leq x \leq 1.8\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: \(1.8\) is approximately where the maximum occurs, not the full domain. The answer is \(0 \leq x \leq 3.8\).</p>
</td>
</tr>
<tr>
<td>
<p>\(0 \leq x \leq 60\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This describes the range. The answer is \(0 \leq x \leq 3.8\).</p>
</td>
</tr>
<tr>
<td>
<p>\(0 \leq x \leq 3.8\)</p>
</td>
<td>
<p>That’s correct! Check yourself: The time in seconds can be between \(0\) and almost \(4\) seconds.</p>
</td>
</tr>
<tr>
<td>
<p>\(x \geq 0\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: Not all values above \(0\) are in the domain. The answer is \(0 \leq x \leq 3.8\).</p>
</td>
</tr>
</tbody>
</table>
<br>
<!--END SELF CHECK INTRO BEFORE Tables -->
<h3>7.7.3: Additional Resources</h3>
<p>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.</p>
<h4>The Meaning of Quadratic Characteristics</h4>
<p>The height in feet of an object \(t\) seconds after being dropped is graphed below: \(h(t)=576-16t^2\).</p>
<p><img alt="<p>Graph of the quadratic function h(t)=576 - 16t^2 on a coordinate plane, origin O. Horizontal axis scale negative 8 to 8 by 4’s, labeled “time (seconds)”. Vertical axis scale 0 to 800 by 200’s, labeled “height (feet)”. Some of the points of this function are (negative 6 comma 0), (negative 4 comma 320) and (negative 2 comma 512) to a maximum at (0 comma 576) then decreasing through (2 comma 512), (4 comma 320) and (6 comma 0).</p>" height="287" src="https://k12.openstax.org/contents/raise/resources/22710f1e7cfc5f8d5c2b14fc760a640611ae9322" width="262"></p>
<p>Find the domain of the function.</p>
<ul>
<li> The <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> domain</span></strong> is the \(t\) values, \(0 \leq t \leq 6\) or only numbers from \(0\) up through \(6\). Negative values of time don’t make sense here, so the part of the graph to the left side of the vertical axis has no meaning. The object hits the ground \(6\) seconds after being dropped, so values greater than \(6\) are not meaningful. </li>
</ul>
<p>Find the range of the function.</p>
<ul>
<li>The <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> range</span></strong> of the height of the function is from the initial height down to the ground. In this case, since the object is being dropped from a height of 576 feet and falls to the ground, the range of the function is [0, 576]. This means that the object's height will be between 0 feet (when it hits the ground) and 576 feet (when it is initially dropped).</li>
</ul>
<p>Find the vertex and describe its meaning.</p>
<ul>
<li> The <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> vertex (of a graph)</span></strong> is the peak of the parabola, or the maximum, at the point \((0,576)\). It tells us the time when the height is the greatest. </li>
</ul>
<p>Find the zeros and describe their meaning.</p>
<ul>
<li> The <strong><span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip"> zero of the function</span></strong> is \(6\). It tells us that the height of the object is \(0\) feet at \(6\) seconds after being dropped, or the time when the object hits the ground. </li>
</ul>
<li> The zero of the function is \(6\). It tells us that the height of the object is \(0\) feet at \(6\) seconds after being dropped, or the time when the object hits the ground. </li>
<h4>Try It: The Meaning of Quadratic Characteristics</h4>
<p>For questions 1 - 2, use the following graph.</p>
<blockquote>
<img height="290" src="https://k12.openstax.org/contents/raise/resources/fc0355928478dd058a1d518ea62a6f20d2dcb493" width="340">
</blockquote>
<ol class="os-raise-noindent">
<li>Describe the zeros of graph.<br>
<p><strong>Answer: </strong> Your answer may vary, but here is a sample.</p>
<p>\(x=0\), \(x=16\)</p>
</li>
<li>Describe what the zeros mean for the graph.<br>
<p><strong>Answer: </strong> Your answer may vary, but here is a sample.</p>
<p> Looking at the label on the \(x\)-axis, this value means that at 0 seconds and 16 seconds, the height is 0 feet.</p>
</li>
</ol>