Linear Functions, Slope, and Applications Section 1.3 Linear Functions, Slope, and Applications Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives Determine the slope of a line given two points on the line. Solve applied problems involving slope. 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 = 0, the function is a constant function f (x) = b. If m = 1 and b = 0, the function is the identity function f (x) = x.

Examples Identity Function Linear Function y = mx + b y = 1•x + 0 or y = x Linear Function y = mx + b

Examples Not a Function Constant Function y = 0•x + b or y = b Vertical line: x = a Constant Function y = 0•x + b or y = b

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. y =  2 x =  2

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 Positive—line slants up from left to right Negative—line slants down from left to right

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 2 5/2 2 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 4 ft for every horizontal distance of 100 ft.

Example Construction laws regarding access ramps for the disabled state that every vertical rise of 1 ft requires a horizontal run of 12 ft. What is the grade, or slope, of such a ramp? The grade, or slope, of the ramp is 8.3%.

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 percent of travel bookings online has increased from 6% in 1999 to 55% in 2007. The graph below illustrates this trend. Find the average rate of change in the percent of travel bookings made online from 1999 to 2007.

Example The coordinates of the two points on the graph are (1999, 6%) and (2007, 55%). The average rate of change over the 8-yr period was an increase of per year.

Slope-Intercept Equation 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 y = – 0.25x – 3.8. y = – 0.25x – 3.8 Slope = –0.25; y-intercept = (0, –3.8)

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 An anthropologist can use linear functions to estimate the height of a male or a female, given the length of the humerus, the bone from the elbow to the shoulder. The height, in centimeters, of an adult male with a humerus of length x, in centimeters, is given by the function The height, in centimeters, of an adult female with a humerus of length x is given by the function . A 26-cm humerus was uncovered in a ruins. Assuming it was from a female, how tall was she?

Example(cont) We substitute into the function: Thus, the female was 142.98 cm tall.