EXAMPLE 3 Approximate a best-fitting line

The table shows the number y(in thousands) of alternative-fueledvehicles in use in the United States x yearsafter 1997. Approximate the best-fittingline for the data.

x 0 1 2 3 4 5 6 7

y 280 295 322 395 425 471 511 548

Alternative-fueled Vehicles

Approximate a best-fitting line

EXAMPLE 3

SOLUTION

STEP 1

Draw a scatter plot of the data.

STEP 2

Sketch the line that appears tobest fit the data. One possibility is shown.

Approximate a best-fitting line

EXAMPLE 3

STEP 3

Choose two points that appear to lie on the line. For theline shown, you might choose (1, 300), which is not anoriginal data point, and (7, 548), which is an original datapoint.

STEP 4

Write an equation of the line. First find the slope usingthe points (1, 300) and (7, 548).

248 6

m = = 41.3548 – 300

7 – 1

Approximate a best-fitting line

EXAMPLE 3

Use point-slope form to write the equation. Choose(x1, y1) = (1, 300).

y – y1 = m(x – x1) Point-slope form

y – 300 = 41.3(x – 1) Substitute for m, x1, and y1.

Simplify.y 41.3x + 259

ANSWER

An approximation of the best-fitting line is y = 41.3x + 259.

EXAMPLE 4 Use a line of fit to make a prediction

Use the equation of the line of fit from Example 3 topredict the number of alternative-fueled vehicles in usein the United States in 2010.

SOLUTION

Because 2010 is 13 years after 1997, substitute 13 for xin the equation from Example 3.

y = 41.3x + 259 = 41.3(13) + 259 796

Use a line of fit to make a prediction

EXAMPLE 4

ANSWER

You can predict that there will be about 796,000 alternative-fueled vehicles in use in the United States in 2010.

Use a graphing calculator to find a best-fitting line

EXAMPLE 5

Use the linear regression feature on a graphing calculatorto find an equation of the best-fitting line for the data inExample 3.

SOLUTION

STEP 1

Enter the data into two lists.Press and then select Edit.Enter years since 1997 in L1 andnumber of alternative-fueled vehicles in L2.

Use a graphing calculator to find a best-fitting line

EXAMPLE 5

STEP 2

Find an equation of the best-fitting (linear regression) line. Press choose the CALC menu, and select LinReg(ax + b). The equationcan be rounded to y = 40.9x + 263.

Use a graphing calculator to find a best-fitting line

EXAMPLE 5

STEP 3Make a scatter plot of the data pairsto see how well the regression equation models the data. Press [STAT PLOT]to set up your plot. Then select an appropriate windowfor the graph.

Use a graphing calculator to find a best-fitting line

EXAMPLE 5

STEP 4

Graph the regression equationwith the scatter plot by entering the equation y = 40.9x + 263. The graph (displayed in the window 0 ≤ x ≤ 8 and 200 ≤ y ≤ 600) shows that the line fitsthe data well.

An equation of the best-fitting line is y = 40.9x + 263.

ANSWER

GUIDED PRACTICE for Examples 3, 4 and 5

4. OIL PRODUCTION: The table shows the U.S. daily oil production y (in thousands of barrels) x years after 1994.

a. Approximate the best-fitting line for the data.

ANSWER y = –130x + 6710

GUIDED PRACTICE for Examples 3, 4 and 5

4. OIL PRODUCTION: The table shows the U.S. daily oil production y (in thousands of barrels) x years after 1994.

b. Use your equation from part (a) to predict the daily oil production in 2009.

ANSWER 4760 gal

GUIDED PRACTICE for Examples 3, 4 and 5

4. OIL PRODUCTION: The table shows the U.S. daily oil production y (in thousands of barrels) x years after 1994.

c. Use a graphing calculator to find and graph an equation of the best-fitting line. Repeat the prediction from part (b) using this equation.

GUIDED PRACTICE for Examples 3, 4 and 5

ANSWER