Linear Functions, Slope, and Applications Section 1.3 Linear Functions, Slope, and Applications

Objectives Determine the slope of a line given two points on the line. Solve applied problems involving slope, or average rate of change. Find the slope and the y-intercept of a line given the equation y = mx + b, or f (x) = mx + b. Graph a linear equation using the slope and the y-intercept. Solve applied problems involving linear functions.

Linear Functions A function f is a linear function if it can be written as f (x) = mx + b, where m and b are constants. If m = 1 and b = 0, the function is the identity function f (x) = x. If m = 0, the function is a constant function f (x) = b.

Examples Linear function: y = mx + b Identity function: y = 1 · x + 0, or y = x

Examples Constant function: Vertical line: x = a y = 0 · x + b, or y = b (Horizontal line) Vertical line: x = a (not a function)

Horizontal and Vertical Lines Horizontal lines are given by equations of the type y = b or f(x) = b. (They are functions.) Vertical lines are given by equations of the type x = a. (They are not functions.)

Slope The slope m of a line containing the points (x1, y1) and (x2, y2) is given by

Example Graph the function and determine its slope. Calculate two ordered pairs, plot the points, graph the function, and determine its slope.

Types of Slopes If a line slants up from left to right, the line has a positive slope. If a line slants down from left to right, the line has a negative slope.

Horizontal Lines If a line is horizontal, the change in y for any two points is 0 and the change in x is nonzero. Thus a horizontal line has slope 0.

Vertical Lines If a line is vertical, the change in y for any two points is nonzero and the change in x is 0. Thus the slope is not defined because we cannot divide by 0.

Example Graph each linear equation and determine its slope. a. x = –2 Choose any number for y ; x must be –2. x y 2 2 Vertical line 2 units to the left of the y-axis. Slope is not defined. Not the graph of a function.

Example (continued) Graph each linear equation and determine its slope. b. Choose any number for x ; y must be 5/2. x y 5/2 −3 1 Horizontal line 5/2 units above the x-axis. Slope 0. The graph is that of a constant function.

Applications of Slope The grade of a road is a number expressed as a percent that tells how steep a road is on a hill or mountain. A 4% grade means the road rises (or falls) 4 ft for every horizontal distance of 100 ft.

Example A federal law states that every vertical rise of 1 ft requires a horizontal run of 12 ft. Find the grade of the curb ramp show in the following figure. The grade, or slope, is given by

Average Rate of Change Slope can also be considered as an average rate of change. To find the average rate of change between any two data points on a graph, we determine the slope of the line that passes through the two points.

Example The number of people participating in the federal Supplemental Nutrition Assistance Program has increased from 17.2 million in 2000 to 47.6 million in 2013. The graph on the following slide illustrates this upward trend. Find the average rate of change in the number of people using food stamps from 2000 to 2013.

Example

Example The coordinates of the two points on the graph are (2000, 17.2) and (2013, 47.6). The average rate of change over the 13-yr period was an increase of 2.3 million participants per year.

Slope-Intercept Equations of Lines The linear function f given by f (x) = mx + b is written in slope-intercept form. The graph of an equation in this form is a straight line parallel to f (x) = mx. The constant m is called the slope, and the y-intercept is (0, b).

Example Find the slope and y-intercept of the line with equation

Example Find the slope and y-intercept of the line with equation 3x – 6y  7 = 0. We solve for y: Thus, the slope is and the y-intercept is

Example Graph The equation is in slope-intercept form, y = mx + b. The y-intercept is (0, 4). Plot this point, then use the slope to locate a second point.

Example There is no proven way to predict a child’s adult height, but a linear function can be used to estimate it, given the sum of the heights of the child’s parents. The adult height M, in inches, of a male child whose parents’ combined height is x, in inches, can be estimated with the function The adult height F, in inches, of a female child whose parents’ combined height is x, in inches can be estimated with the function

Example (cont.) Estimate the height of a female child whose parents’ combined height is 135 in. Solution: We substitute in the function: Thus we can estimate the adult height of the female child is 65in., or 5 ft 5 in.