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<h4>Activity (15 minutes)</h4>
<p>In this activity, students continue to familiarize themselves with the process of entering information into Desmos. They use quadratic regression on a data set to find the equation of a curve of best fit. </p>
<p>In the next activity, they will use a similar curve of best fit to make predictions about a situation.</p>
<h4>Launch</h4>
<p>Arrange students in pairs or groups, if necessary, so they have access to Desmos. If possible, allow students to work individually so that each understands the steps necessary to find a curve of best fit.</p>
<p>Read the activity introduction as a class. Ask students if they remember finding the line of best fit of a data set. Introduce the new vocabulary words. Tell students they have already used linear regression in a previous lesson, and they will now use quadratic regression.</p>
<p>Allow students to open the graphed data set below and answer question 2.</p>
<div class="os-raise-ib-desmos-gc" data-bottom="-50" data-equations='["y_1~ax_1^2+bx_1+c"]' data-left="-50" data-right="50" data-schema-version="1.0" data-tables='[[{"variable": "x_1", "values": [0.5,1,2,2.5,3,3.5]}, {"variable": "y_1", "values": [20.2,24.1,28.8,31.6,33.1,34.2]}]]' data-top="50"></div>
<p>If possible, display the graph for the class to see. Read each bullet point in question 3 before entering the equation. Demonstrate the keys pressed to accurately enter the equation, \(y_1 \sim ax_1^2+bx_1+ c\). Then, if possible, instruct students to enter the equation on their own screens.</p>
<p>Allow students to complete questions 5–9 on their own. Remind them that exponents may be entered using the <q>^</q> symbol.</p>
<p>For English language learners, display the keys that need to be pressed to enter the equation for the curve of best fit:</p>
<table class="os-raise-textheavytable">
<thead>
<tr><th scope="col"></th>
<th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th><th scope="col"></th>
</tr></thead>
<tbody>
<tr>
<td>
y
</td>
<td>
1
</td>
<td>
~
</td>
<td>
a
</td>
<td>
x
</td>
<td>
1
</td>
<td>
^
</td>
<td>
2
</td>
<td>
+
</td>
<td>
b
</td>
<td>
x
</td>
<td>
1
</td>
<td>
+
</td>
<td>
c
</td>
</tr>
</tbody>
</table>
<br>
<p>Reinforce that the <q>Shift</q> button will need to be held down first when pressing the <q>~</q> symbol.</p>
<h4>Student Activity</h4>
<p>In a previous lesson, you learned how to find the line of best fit. A line of best fit is a line on a graph that best approximates the equation of the points in a data set or scatter plot.</p>
<p>In this activity, we will learn how to find the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">curve of best fit</span>, which is the curve modeled by the equation that best approximates the points in a data set or scatter plot. Since our data set is based on a quadratic function, we must use a curve instead of a line. This process, called <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">quadratic regression</span>, is the process of determining the equation of a parabola that best fits a data set.</p>
<ol class="os-raise-noindent">
<li>Examine the data set and graph given in the Desmos graphing tool below.</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-bottom="-10" data-left="-1" data-right="12" data-schema-version="1.0" data-tables='[[{"variable": "x_1", "values": [0.5,1,2,2.5,3,3.5]}, {"variable": "y_1", "values": [20.2,24.1,28.8,31.6,33.1,34.2]}]]' data-top="60"></div>
<ol class="os-raise-noindent" start="2">
<li>Does the data appear to form a line or a curve? To answer question 2, allow students to examine the graphed data set. (Students were provided access to Desmos: graphed data set)</li>
</ol>
<p><strong>Answer:</strong> They appear to be the start of a curve.</p>
<ol class="os-raise-noindent" start="3">
<li>We will use graphing technology to find a curve of best fit. Click the pointer below the table into entry box #2 to enter a new function. The cursor should start blinking. Enter the equation:</li>
</ol>
<p> \(y_1 \sim ax_1^2+bx_1+ c\) </p>
<ul class="os-raise-noindent">
<li> Notice that when you type the <q>1</q> after \(y\) or \(x\), Desmos automatically places it as subscript. </li>
</ul>
<ul class="os-raise-noindent">
<li> Also notice that we are not using an equal sign. The symbol <q>~</q> means approximately and tells Desmos to approximate a line or curve of best fit. It should be in the top-left corner of your keyboard, and you might press it using the Shift button. If you are using the Desmos on-screen keyboard, it will be on the ABC keyboard. </li>
</ul>
<ul class="os-raise-noindent">
<li> After pressing the subscript 1 after the \(y\) and the first \(x\), use the <q>^</q> symbol to tell Desmos you would like to enter an exponent. </li>
</ul>
<ol class="os-raise-noindent" start="4">
<li>Once you have entered the curve of best fit, you can press Enter. The curve of best fit will appear. Your screen will look like this:</li>
</ol>
<div class="os-raise-ib-desmos-gc" data-bottom="-10" data-equations='["y_1~ax_1^2+bx_1+c"]' data-left="-1" data-right="12" data-schema-version="1.0" data-tables='[[{"variable": "x_1", "values": [0.5,1,2,2.5,3,3.5]}, {"variable": "y_1", "values": [20.2,24.1,28.8,31.6,33.1,34.2]}]]' data-top="60"></div>
<p>Look in the lower-left corner of the screen. We will use the \(a\), \(b\), and \(c\) values to construct the equation for our curve of best fit.</p>
<ol class="os-raise-noindent" start="5">
<li>What is the value of \(a\) rounded to the nearest hundredth?</li>
</ol>
<p><strong>Answer:</strong> -0.86</p>
<ol class="os-raise-noindent" start="6">
<li>What is the value of \(b\) rounded to the nearest hundredth?</li>
</ol>
<p><strong>Answer:</strong> 8.09</p>
<ol class="os-raise-noindent" start="7">
<li>What is the value of \(c\) rounded to the nearest hundredth?</li>
</ol>
<p><strong>Answer:</strong> 16.53</p>
<ol class="os-raise-noindent" start="8">
<li>Write an equation in the form \(y = ax^2+ bx + c\) for the curve of best fit using the values you found.</li>
</ol>
<p><strong>Answer:</strong> \(y = -0.86x^2 + 8.09x + 16.53\)</p>
<p>This is the equation for your curve of best fit.</p>
<p>You may notice the \(R^2\) value. The closer this value is to 1, the better your curve of best fit matches the data set. The \(R^2\) value for this data set is 0.9972. Since this value is relatively close to 1, this curve of best fit is a reasonably accurate match to the data set. </p>
<h4>Activity Synthesis</h4>
<p>Reinforce the idea that the curve of best fit does not precisely align with each point in the data set. In other words, no point in the data set is guaranteed to fall exactly onto the curve of best fit. </p>
<p>Inform students that while the \(R^2\) value is an important measurement of how closely the curve of best fit matches the data set, we will not be using it in this lesson.</p>
<p>To review key ideas from the activity, discuss questions such as:</p>
<ul class="os-raise-noindent">
<li> <q>If you graph the equation, \(y = -0.86x^2 + 8.09x + 16.53\), into Desmos, how will it compare to the curve of best fit shown on the screen now?</q> (It will be nearly identical, aside from rounding differences, to the curve of best fit that was derived using the equation, \(y_1 \sim ax_1^2+bx_1+ c\).) </li>
<li> <q>What do you think would happen if you changed one of the values in the data set?</q> (Changing the value of a point in the data set would change the equation of the curve of best fit. It would no longer be \(y = -0.86x^2 + 8.09x + 16.53\).) </li>
</ul>
<p>Demonstrate the impact that changing a value in one point of the data set, even slightly, will have on the equation of the curve of best fit. (Use one of the prior data sets that students were provided access to in Desmos.)</p>
<h3> 8.12.2: 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>Find the equation of the curve of best fit. Round each coefficient to the nearest hundredth.</p>
<p>(Students were given access to Desmos. Here is the <a href="https://k12.openstax.org/contents/raise/resources/424f5834953dd9e10a9c0752ab21d11aa211ba22" target="_blank">dataset</a> provided.)</p>
<p>Which is the equation of the curve of best fit?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">ANSWERS</th>
<th scope="col">FEEDBACK</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(y = -0.92x^2+ 4.25x + 8.36\)
</td>
<td>
Incorrect. Let's try again a different way: Did you enter \(y_1 \sim ax_1^2+bx_1+ c\) to find the curve of best fit? The answer is \(y = -0.94x^2+ 4.51x + 16.96\).
</td>
</tr>
<tr>
<td>
\(y = -0.97x^2+ 7.62x + 19.34\)
</td>
<td>
Incorrect. Let's try again a different way: Did you enter \(y_1 \sim ax_1^2+bx_1+ c\) to find the curve of best fit? The answer is \(y = -0.94x^2+ 4.51x + 16.96\).
</td>
</tr>
<tr>
<td>
\(y = -0.94x^2+ 4.51x + 16.96\)
</td>
<td>
That's correct! Check yourself: The graph of the curve lines up with the data set as closely as possible.
</td>
</tr>
<tr>
<td>
\(y = -0.91x^2 + 2.23x + 16.42\)
</td>
<td>
Incorrect. Let's try again a different way: Did you enter \(y_1 \sim ax_1^2+bx_1+ c\) to find the curve of best fit? The answer is \(y = -0.94x^2+ 4.51x + 16.96\).
</td>
</tr>
</tbody>
</table>
<br>
<h3>8.12.2: 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>Finding the Curve of Best Fit </h4>
<p>Let's find the curve of best fit for a given data set.</p>
<p>Click on the Desmos link to open a graph.</p>
<p>
<p>(Students were provided access to Desmos with a set \(x\)-axis of \(-10 < x < 20\) and a set \(y\)-axis of \(-25 < y < 225)\).</p>
</p>
<p><strong>Step 1</strong> - Click the + symbol in the top left and select a table.</p>
<p>Enter the data set as shown. Press Enter.</p>
<table class="os-raise-skinnytable">
<thead>
<tr>
<th scope="col">
\(x_1\)
</th>
<th scope="col">
\(y_1\)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
1
</td>
<td>
99.7
</td>
</tr>
<tr>
<td>
2
</td>
<td>
130
</td>
</tr>
<tr>
<td>
3
</td>
<td>
149
</td>
</tr>
<tr>
<td>
4
</td>
<td>
172
</td>
</tr>
<tr>
<td>
9
</td>
<td>
175
</td>
</tr>
<tr>
<td>
11
</td>
<td>
140
</td>
</tr>
</tbody>
</table>
<br>
<p>The data set will populate the graph as shown here.</p>
<p><img height="295" src="https://k12.openstax.org/contents/raise/resources/9a495e305858f322af933bda428076d4e00b1495" width="624"></p>
<p>Step 2 <br>
On the left side, click in box #2 for a new item. Enter the instruction line below to generate a curve of best fit.</p>
<p>\(y_1 \sim ax_1^2+bx_1+ c\)</p>
<ul class="os-raise-noindent">
<li> Remember from the activity that Desmos will automatically place the 1 as subscript after an \(x\) or \(y\).</li>
</ul>
<ul class="os-raise-noindent">
<li> Don't forget to use the symbol <q>~</q> that means approximately. </li>
</ul>
<ul class="os-raise-noindent">
<li> Use the <q>^</q> symbol to enter an exponent. </li>
</ul>
<p>As soon as you enter the instruction line, Desmos will generate the curve of best fit.</p>
<p><strong>Step 3</strong> - Use the coefficients generated to write the equation for the curve of best fit.</p>
<p><img alt="OUTPUT SUMMARY FOR A QUADRATIC REGRESSION. R-SQUARED IS 0.9958. A, B, AND C ARE NEGATIVE 2.75569, 37.0352, AND 65.6605, RESPECTIVELY." height="229" src="https://k12.openstax.org/contents/raise/resources/7465a3766f7e69e2ca46d5144c2e8feaeb91c3c7" width="483"></p>
<p>Round each coefficient to the nearest hundredth. </p>
<p>The curve of best fit is \(y = -2.76x^2+ 37.04x + 65.66\).</p>
<h4>Try It: Finding the Curve of Best Fit</h4>
<p>Find the equation of the curve of best fit. Round each coefficient to the nearest hundredth.</p>
<div class="os-raise-ib-desmos-gc" data-bottom="-25" data-left="-10" data-right="20" data-schema-version="1.0" data-tables='[[{"variable": "x_1", "values": [0.5,2,3,4,5]}, {"variable": "y_1", "values": [30,58,93,147,201]}]]' data-top="225"></div>
<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 curve of best fit:</p>
<p>Solve for the curve of best fit by entering \(y_1 \sim ax_1^2+bx_1+ c\). </p>
<p>The curve of best fit is \(y = 6.18x^2+ 4.59x + 25.42\).</p>