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<h4>Activity (20 minutes)</h4>
<p>Previously, students interpreted rules of functions only in terms of the operations performed on the input to lead to the output. In this activity, students analyze functions that relate two quantities in a situation and work to define the relationship between the quantities with a rule. They do so by creating a table of values and generalizing the process of finding one quantity given the other. Students also plot the values in each table to see the graphical representation of the functions.</p>
<p>The mathematical reasoning here is not new. Students have done similar work earlier in the course, when investigating expressions and equations. What is new is seeing these relationships as functions and using function notation to describe them.</p>
<p>Students are likely to graph the functions by plotting the values in the tables and then connecting the points with a curve. As students work on the second set of questions about a perimeter function, which is linear, look for those who relate \(P(l)=2l+6\) to a linear equation, namely \(y=2x+6\), and then graph a line with a vertical intercept of \((0,6)\) and a slope of 2. Invite them to share their thoughts during the whole-class discussion.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Give students a few minutes of quiet time to work on the first set of questions, and then a moment to discuss their responses with their partner. Then, pause for a brief discussion before students proceed to the second set of questions.</p>
<p>Invite students to share their rule for the area function. Some students may have written \(A=s^2\), while others may have written \(A(s)=s^2\). Ask students who wrote each way to explain their reasoning. Highlight explanations that point out that \(A\) is the name of the function and that function notation requires specifying the input, which is \(s\).</p>
<p>Clarify that in the past, we may have used a variable like \(A\) to represent the area, but in this case, \(A\) is used to name a function to help us talk about its input and output. If we wish to also use a variable to represent the output of this function (instead of using function notation), it would be helpful to use a different letter.</p>
<h4>Student Activity</h4>
<ol class="os-raise-noindent">
<li>A square that has a side length of 9 cm has an area of 81 \(cm^2\). The relationship between the side length and the area of the square is a function.<br>
<br>
<p>For questions a-e use the table by answering each question.</p>
</li>
</ol>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
<p>side length (cm)</p>
</th>
<th scope="col">
<p>area (cm2)</p>
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>1</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>2</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>4</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>6</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>\(s\)</p>
</td>
<td><br>
<br>
</td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li>What is the area in \(cm^2\) when the side length is \(1 cm\)?</li>
<p><strong>Answer: </strong> 1</p>
<li>What is the area in \(cm^2\) when the side length is \(2 cm\)?</li>
<p><strong>Answer: </strong> 4</p>
<li>What is the area in \(cm^2\) when the side length is \(4 cm\)?</li>
<p><strong>Answer: </strong> 16</p>
<li>What is the area in \(cm^2\) when the side length is \(6 cm\)?</li>
<p><strong>Answer: </strong> 36</p>
<li>In your math notebook, write a rule for a function, A, that gives the area of the square in cm2 when the side length is s cm, then click solution to compare your answer.</li>
<p><strong>Answer: </strong> \(A(s)= s^2\)</p>
<li>What does \(A(2)\) represent in this situation? What is its value? Write your answers in your math notebook, then click solution to compare your answer.</li>
<p><strong>Answer: </strong> \(A(2)\) is the area of the square in cm2 when the side length is 2 cm. Its value is 4.</p>
<li>Sketch a graph of this function. (Students were provided access to Desmos) </li>
<p><strong>Answer: </strong> <br>
<img alt height="199" role="presentation" src="https://k12.openstax.org/contents/raise/resources/09ea7f96b3cf45cb99f26dcf599536e434d70294" width="315"> <br>
</p>
</ol>
</ol>
<ol class="os-raise-noindent" start="2">
<li>A roll of paper that is 3 feet wide can be cut to any length.
<ol class="os-raise-noindent" type="a">
<li>If we cut a length of 2.5 feet, what is the perimeter of the paper?</li>
<p> <strong>Answer: </strong> 11 feet</p>
</ol>
</li>
</ol>
<p><img alt="Roll of paper, slightly unrolled. Its width is 3 feet." src="https://k12.openstax.org/contents/raise/resources/78400b22278b2533130a0fdd72c08cda0db81912" width="300"><br></p>
<p>For questions b-f, keep in mind that the roll of paper is 3 feet wide. Complete the table by answering each question.</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
side length (feet)
</th>
<th scope="col">
perimeter (feet)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>1</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>2</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>6.3</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>11</p>
</td>
<td><br></td>
</tr>
<tr>
<td>
<p>\(l\)</p>
</td>
<td><br></td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>If we cut a length of 1 foot, what is the perimeter of the paper to the nearest whole foot? </li>
<p> <strong>Answer: </strong> 8</p>
<li>To the nearest whole foot, what is the perimeter when the side length is 2 feet?</li>
<p> <strong>Answer: </strong> 10</p>
<li>What is the exact perimeter, in feet, when the side length is 6.3 feet? </li>
<p> <strong>Answer: </strong> 18.6</p>
<li>What is the perimeter, to the nearest whole foot, when the side length is 11 feet? </li>
<p> <strong>Answer: </strong> 28</p>
<li>Write a rule for a function, <em>P</em>, that gives the perimeter of the paper in feet when the side length in feet is <em>l</em>.</li>
<p> <strong>Answer: </strong> \(P(l)=6+2l\)</p>
<li>What does P(11) represent in this situation? What is its value?</li>
<p> <strong>Answer: </strong> \(P(11)\) is the perimeter of the paper in feet when the paper is cut at a length of 11 feet. Its value is 28</p>
<li>Sketch a graph of this function. (Students were provided access to Desmos)</li>
<p> <strong>Answer: </strong> <br><img src="https://k12.openstax.org/contents/raise/resources/6f92f382ce0e85d4be2f9984851c75e66db211ed"></p>
</ol>
</ol>
<h4>Video: Writing Function Rules</h4>
<p>Watch the following video to learn more about writing rules with function notation.</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/0b0b2954ac7af8795e4139744d9614b45807b6d2">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/4bc18e6127fd40ef370bb23db8e3080523984086" srclang="en_us">https://k12.openstax.org/contents/raise/resources/0b0b2954ac7af8795e4139744d9614b45807b6d2
</video></div>
</div>
<br>
<br>
<h4>Anticipated Misconceptions</h4>
<p>If students struggle to graph the functions, suggest that they use the coordinate pairs in the tables to help them.</p>
<h4>Activity Synthesis</h4>
<p>Select students to share the rule they wrote for the perimeter function (from the second set of questions) and how they determined the rule. Students may have written expressions of different forms for \(P(l)\):</p>
<p>\(l+l+3+3\)</p>
<p>\(2l+2(3)\)</p>
<p>\(2(l+3)\)</p>
<p>\(6+2l\)</p>
<p>Record and display the variations for all to see and discuss whether they all give the value of \(P(l)\). Ask students to explain how they know these expressions are equivalent and define the same function.</p>
<p>Next, discuss how students sketched the graph of the function. If no students made a connection between the slope and vertical intercept of the graph of \(P\) to the parameters in their equation, ask them about it. For example, display the graph of \(P\) and ask students to use it to write an equation for the line.</p>
<br>
<h4>4.4.3: Self Check</h4>
<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>Which function rule can describe the table below?</p>
<table class="os-raise-skinnytable">
<thead>
<tr>
<th scope="col">
\(x\)
</th>
<th scope="col">
\(y\)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>1</p>
</td>
<td>
<p>3</p>
</td>
</tr>
<tr>
<td>
<p>2</p>
</td>
<td>
<p>5</p>
</td>
</tr>
<tr>
<td>
<p>5</p>
</td>
<td>
<p>11</p>
</td>
</tr>
</tbody>
</table>
<br>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">ANSWERS</th>
<th scope="col">FEEDBACK</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(f(x)=x+3\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: When 1 is substituted for \(x\), \((1)+3=4\), not 3. This function does not work for the values. The answer is \(f(x)=2x+1\).</p>
</td>
</tr>
<tr>
<td>
<p>\(f(x)=2x+1\)</p>
</td>
<td>
<p>That’s correct! Check yourself: When \(x =1\), \(2(1)+1=3\), when \(x=2\), \(2(2)+1=5\), and when \(x=5\), \(2(5)+1=11\).</p>
</td>
</tr>
<tr>
<td>
<p>\(f(x)=2x+2\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: When 1 is substituted for \(x\), \(2(1)+2=4\), not 3. This function does not work for all values. The answer is \(f(x)=2x+1\).</p>
</td>
</tr>
<tr>
<td>
<p>\(f(x)=x+2\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: While this rule works for the first row of the table, when \(x = 2\), \((2)+2=4\), not 5, so the rule does not work for all values in the table. The answer is \(f(x)=2x+1\).</p>
</td>
</tr>
</tbody>
</table>
<br>
<h3>4.4.3: 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>Write the Function Rule from a Table</h4>
<p>Kelly can tune 6 cars in 3 hours. The table below is completed to show this relationship. The last column has the output for any \(x\). This is the rule of the function.</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">
Time spent tuning cars (\(x\))
</th>
<td>
2
</td>
<td>
3
</td>
<td>
6
</td>
<td>
7
</td>
<td>
\(x\)
</td>
</tr>
<tr>
<th scope="row">
Number of cars tuned up (\(y\))
</th>
<td>
<p>4</p>
</td>
<td>
<p>6</p>
</td>
<td>
<p>12</p>
</td>
<td>
<p>14</p>
</td>
<td>
<p>\(2x\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>Kelly can tune 6 cars in 3 hours or 2 cars in 1 hour, so the function rule is \(f(x)=2x\).</p>
<h4>Try It: Write the Function Rule from a Table</h4>
<p>Determine the rule in function notation for the table below:</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">
\(x\) (input)
</th>
<td>
1
</td>
<td>
2
</td>
<td>
3
</td>
<td>
4
</td>
<td>
\(x\)
</td>
</tr>
<tr>
<th scope="row">
\(y\) (output)
</th>
<td>
5
</td>
<td>
8
</td>
<td>
11
</td>
<td>
14
</td>
<td>
?
</td>
</tr>
</tbody>
</table>
<br>
<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 rule in function notation:</p>
<p>In the table, every \(x\) is multiplied by 3, and then 2 is added. The rule is:</p>
<p>\(f(x) = 3x + 2\)</p>
<br>