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9e617b29-706b-4f9d-b0fb-207e30c62d81.html
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<h4>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>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e626bd5f-3e54-4ebd-a281-4634b4be71bc" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Examine the data. Do they appear to you to form a line or a curve?</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>They appear to be the start of a curve.</p>
</div>
</div>
<!--Interaction End -->
<br>
<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 class="os-raise-noindent"> \(y_1 \sim ax_1^2+bx_1+ c\) </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>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="9da255ba-eb1d-4c8f-9990-946407fb0eaa" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q6-->
<div class="os-raise-ib-pset-problem" data-content-id="df117ed1-5d4f-435a-8348-c5ae806d81dd" data-problem-comparator="float" data-problem-type="input" data-solution="-0.86">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="5">
<li>What is the value of \(a\) rounded to the nearest hundredth?</li>
</ol>
<p>Enter the value of \(a\).</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(– 0.86\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is \(– 0.86\) </p>
</div>
</div>
<!--Q2-->
<div class="os-raise-ib-pset-problem" data-content-id="48824ada-f643-4d81-b691-760b82408396" data-problem-comparator="float" data-problem-type="input" data-solution="8.09">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="6">
<li>What is the value of \(b\) rounded to the nearest hundredth?</li>
</ol>
<p>Enter the value of \(b\).</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(8.09\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is \(8.09\)</p>
</div>
</div>
<!--Q3-->
<div class="os-raise-ib-pset-problem" data-content-id="5a9d0b4a-3348-4e1f-9f9f-dd1a56334c82" data-problem-comparator="float" data-problem-type="input" data-solution="16.53">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="7">
<li>What is the value of \(c\) rounded to the nearest hundredth?</li>
</ol>
<p>Enter the value of \(c\).</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! \(16.53\)</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Nice try! The correct answer is \(16.53\)</p>
</div>
</div>
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response"> </div>
</div>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="99e7fbcb-38eb-4eca-bc3d-a146b3725b4e" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<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>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter an equation.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(y = -0.86x^2 + 8.09x + 16.53\)</p>
<p>This is the equation for your curve of best fit.</p>
</div>
</div>
<!--Interaction End -->
<br>
<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>