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Algebra 1

8.12.2 Finding the Curve of Best Fit

Algebra 18.12.2 Finding the Curve of Best Fit

8.12.2 Finding the Curve of Best Fit

8.12.2 • Finding the Curve of Best Fit

Activity

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.

In this activity, we will learn how to find the curve of best fit, 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 quadratic regression, is the process of determining the equation of a parabola that best fits a data set.

1.

Examine the data set and graph given in the Desmos graphing tool below.

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2.

Examine the data. Do they appear to you to form a line or a curve?

3.

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:

y 1 ~ a x 1 2 + b x 1 + c y 1 ~ a x 1 2 + b x 1 + c

  • Notice that when you type the 1 after y y or x x , Desmos automatically places it as subscript.
  • Also notice that we are not using an equal sign. The symbol ~ 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.
  • After pressing the subscript 1 after the y y and the first x x , use the ^ symbol to tell Desmos you would like to enter an exponent.
4.
  1. 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:
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Look in the lower-left corner of the screen. We will use the a a , b b , and c c values to construct the equation for our curve of best fit.

5.

Enter the value of a a .

What is the value of a a rounded to the nearest hundredth?

6.

Enter the value of b b .

What is the value of b b rounded to the nearest hundredth?

7.

Enter the value of c c .

What is the value of c c rounded to the nearest hundredth?

8.

Write an equation in the form y = a x 2 + b x + c y = a x 2 + b x + c for the curve of best fit using the values you found.

You may notice the R 2 R 2 value. The closer this value is to 1, the better your curve of best fit matches the data set. The R 2 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.

Self Check

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Find the equation of the curve of best fit. Round each coefficient to the nearest hundredth.

Which is the equation of the curve of best fit?

  1. y = − 0.91 x 2 + 2.23 x + 16.42
  2. y = − 0.94 x 2 + 4.51 x + 16.96
  3. y = − 0.97 x 2 + 7.62 x + 19.34
  4. y = − 0.92 x 2 + 4.25 x + 8.36

Additional Resources

Finding the Curve of Best Fit

Let's find the curve of best fit for a given data set.

Use the Desmos graphing tool to determine the equation.

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Step 1 - Click the + symbol in the top left and select a table.

Enter the data set as shown. Press Enter.

x 1 x 1 y 1 y 1
1 99.7
2 130
3 149
4 172
9 175
11 140

The data set will populate the graph as shown here.

Step 2 - On the left side, click in box #2 for a new item. Enter the instruction line below to generate a curve of best fit.

y 1 ~ a x 1 2 + b x 1 + c y 1 ~ a x 1 2 + b x 1 + c

  • Remember from the activity that Desmos will automatically place the 1 as subscript after an x x or y y .
  • Don't forget to use the symbol ~ that means approximately.
  • Use the ^ symbol to enter an exponent.

As soon as you enter the instruction line, Desmos will generate the curve of best fit.

Step 3 - Use the coefficients generated to write the equation for the curve of best fit.

Round each coefficient to the nearest hundredth.

The curve of best fit is y = − 2.76 x 2 + 37.04 x + 65.66 y = − 2.76 x 2 + 37.04 x + 65.66 .

Try it

Finding the Curve of Best Fit

Find the equation of the curve of best fit. Round each coefficient to the nearest hundredth.

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