Introduction The relationship between two variables can be estimated using a function. The equation can be used to estimate values that are not in the observed data set. To determine which type of equation should be used for a data set, first create a scatter plot of the data. Data that has a linear shape, or can be approximated by a straight line, can be fitted to a linear equation. Points in the data set can be used to find a linear equation that is a good approximation for the data. Only two points are needed to draw a line. 4.2.4: Fitting Linear Functions to Data

Introduction, continued Drawing a line through two data points on the same coordinate plane as the scatter plot helps display how well the line matches the data set. If the line is a good fit for the data, some data points will be above the line and some data points will be below the line. After creating a graphical representation of a line that fits the data, find the equation of this line using the two known points on the line. Use the two known points on the line to calculate the slope and y-intercept of the line. 4.2.4: Fitting Linear Functions to Data

Key Concepts A scatter plot that can be estimated with a linear function will look approximately like a line. A line through two points in the scatter plot can be used to find a linear function that fits the data. If a line is a good fit for a data set, some of the data points will be above the line and some will be below the line. The general equation of a line in point-slope form is y = mx + b, where m is the slope and b is the y-intercept. 4.2.4: Fitting Linear Functions to Data

Key Concepts, continued To find the equation of a line with two known points, calculate the slope and y-intercept of a line through the two points. Slope is the change in y divided by the change in x; a line through the points (x1, y1) and (x2, y2) has a slope of . 4.2.4: Fitting Linear Functions to Data

Key Concepts, continued To find the y-intercept, or b in the equation y = mx + b, replace m with the calculated slope, and replace x and y with values of x and y from a point on the line. Then solve the equation for b. For example, for a line with a slope of 2 containing the point (1, –3), m = 2, y = –3, and x = 1; –3 = (2)(1) + b, and –5 = b. 4.2.4: Fitting Linear Functions to Data

Common Errors/Misconceptions thinking that a line is a good estimate for data that is not linear drawing a line that is not a good fit for the data, and calculating the equation of this line miscalculating the slope of a line using two points on the line incorrectly calculating the y-intercept when finding the equation of a line given a graph 4.2.4: Fitting Linear Functions to Data

Guided Practice Example 1 A weather team records the weather each hour after sunrise one morning in May. The hours after sunrise and the temperature in degrees Fahrenheit are in the table to the right. Hours after sunrise Temperature in ˚F 52 1 53 2 56 3 57 4 60 5 63 6 64 7 67 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued Can the temperature 0–7 hours after sunrise be represented by a linear function? If yes, find the equation of the function. 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued Create a scatter plot of the data. Let the x-axis represent hours after sunrise and the y-axis represent the temperature in degrees Fahrenheit. Temperature (°F) Hours after sunrise 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued Determine if the data can be represented by a linear function. The graph of a linear equation is a line. If the data looks like it could fit a line, then a linear equation could be used to represent the data. The temperatures appear to increase in a line, and a linear equation could be used to represent the data set. 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued Draw a line to estimate the data set. Two points in the data set can be used to draw a line that estimates that data. When the line is drawn, some of the data values should be above the line, and some should be below the line. A line through (2, 56) and (6, 64) looks like a good fit for the data. 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued Temperature (°F) Hours after sunrise 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued Find the equation of the line. The general equation of a line in point-slope form is y = mx + b, where m is the slope, and b is the y-intercept. Find the slope, m, of the line through the two chosen points. The slope is . For any two points (x1, y1) and (x2, y2), the slope is . 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued For the two points (2, 56) and (6, 64), the slope is . Next, find the y-intercept, b. Use the general equation of a line to solve for b. Substitute x and y from a known point on the line, and replace m with the calculated slope. y = mx + b For the point (2, 56): 56 = 2(2) + b; b = 52 4.2.4: Fitting Linear Functions to Data

✔ Guided Practice: Example 1, continued Replace m and b with the calculated values in the general equation of a line. y = 2x + 52 The temperature between 0 and 7 hours after sunrise can be approximated with the equation y = 2x + 52. ✔ 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 1, continued http://walch.com/ei/CAU4L2S4LinFun 4.2.4: Fitting Linear Functions to Data

Guided Practice Example 3 Automated tractors can mow lawns without being driven by a person. A company runs trials using fields of different sizes, and records the amount of time it takes the tractor to mow each field. The field sizes are measured in acres. Acres Time in hours 5 15 7 10 22 17 32.3 18 46.8 20 34 39.6 25 75 30 70 40 112 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued Can the time to mow acres of a field be represented by a linear function? If yes, find the equation of the function. 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued Create a scatter plot of the data. Let the x-axis represent the acres and the y-axis represent the time in hours. Time Acres 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued Determine if the data can be represented by a linear function. The graph of a linear equation is a line. If the data looks like it could fit a line, then a linear equation could be used to represent the data. The temperatures appear to increase in a line, and a linear equation could be used to represent the data set. 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued Draw a line to estimate the data set. Two points in the data set can be used to draw a line that estimates the data. When the line is drawn, some of the data values should be above the line, and some should be below the line. A line through (7, 10) and (40, 112) looks like a good fit for the data. 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued Time Acres 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued Find the equation of the line. The general equation of a line in point-slope form is y = mx + b, where m is the slope, and b is the y-intercept. Find the slope, m, of the line through the two chosen points. The slope is . For any two points (x1, y1) and (x2, y2), the slope is . 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued For the two points (7, 10) and (40, 112), the slope is . Next, find the y-intercept, b. Use the general equation of a line to solve for b. Substitute x and y from a known point on the line, and replace m with the calculated slope. y = mx + b 4.2.4: Fitting Linear Functions to Data

✔ Guided Practice: Example 3, continued For the point (7, 10): 10 = 3.1(7) + b; b = –12 Replace m and b with the calculated values in the general equation of a line. y = 3.1x – 12 The amount of time to mow the acres of a field can be represented using the equation y = 3.1x – 12. ✔ 4.2.4: Fitting Linear Functions to Data

Guided Practice: Example 3, continued http://walch.com/ei/CAU4L2S4LinFunData 4.2.4: Fitting Linear Functions to Data