Lines of Best Fit A line of best fit is a line that comes close to all the points on a scatter plot. Try to draw the line so that about the same number of points are above the line as below the line.

Lines of Best Fit Example 1 Draw a line through that best represents the data. Estimate and plot the coordinates of another point on that line, such as (8, 6). Find the equation of the line.

Example 1 Continued 2 3 1 m = = = 6 – 4 8 – 6 Find the slope. y – y1 = m(x – x1) Use point-slope form. y – 4 = (x – 6) 2 3 Substitute. y – 4 = x – 4 2 3 2 3 y = x + The equation of a line of best fit is . 2 3 y = x +

Example 2 Draw a line through that best represents the data. Estimate and plot the coordinates of another point on that line, such as (10, 10). Find the equation of the line.

Example 2 Continued m = = 10 – 1 10 – 2 9 8 Find the slope. y – y1 = m(x – x1) Use point-slope form. y – 1 = (x – 2) 9 8 Substitute. y – 1 = x – 9 8 4 y = x – 9 8 5 4 The equation of a line of best fit is . y = x – 9 8 5 4

Example 3: Sports Application Lines of Best Fit Example 3: Sports Application Find a line of best fit for the Main Street Elementary annual softball toss. Use the equation of the line to predict the winning distance in 2006. Is it reasonable to make this prediction? Explain. Year 1990 1992 1994 1997 2002 Distance (ft) 98 101 103 106 107 Let 1990 represent year 0. The first point is then (0, 98), and the last point is (12, 107).

Example 3 Continued Draw a line through that best represents the data. Estimate and plot the coordinates of another point on that line, such as (10, 107). Find the equation of the line.

Example 3 Continued m = = 0.8 107 - 103 10 - 5 Find the slope. y – y1 = m(x – x1) Use point-slope form. y – 103 = 0.8(x – 5) Substitute. y – 103 = 0.8x – 4 y = 0.8x + 99 The equation of a line of best fit is y = 0.8x + 99. Since 1990 represents year 0, 2006 represents year 16.

Example 3 Continued y = 0.8(16) + 99 Substitute. y = 12.8 + 99 Add to find the distance. y = 111.8 The equation predicts a winning distance of about 112 feet for the year 2006. A toss of about 112 feet is a reasonable prediction.

Example 4 Predict the winning weight lift in 2010. Year 1990 1995 1997 1998 2000 Lift (lb) 100 120 130 140 170 Let 1990 represent year 0. The first point is then (0, 100), and the last point is (10, 170).

Example 4 Continued Draw a line through the best represents the data. Estimate and plot the coordinates of another point on that line, such as (7, 140). Find the equation of the line. Years since 1990 weight (lb) 100 120 140 160 180 2 4 6 8 10 200

Example 4 Continued m = = 4 140 – 132 7 – 5 Find the slope. y – y1 = m(x – x1) Use point-slope form. y – 132 = 4(x – 5) Substitute. y – 132 = 4x – 20 y = 4x + 112 The equation of a line of best fit is y = 4x + 112. Since 1990 represents year 0, 2010 represents year 20.

Example 4 Continued y = 4(20) + 112 Substitute and add to find the winning weight lift. y = 192 The equation predicts a winning weight lift of about 192 lb for the year 2010. A weight lift of 193 lbs is a reasonable prediction.