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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 Linear Function y = mx + b Identity Function y = 1•x + 0 or y = x Examples Constant Function y = 0•x + b or y = b Not a Function Vertical line: x = a Horizontal and Vertical Lines Horizontal lines are given by Vertical lines are given by equations of the type y = b equations of the type x = a. or f(x) = b. They are They are not functions. functions. x=2 y=2 Slope The slope m of a line containing the points (x1, y1) and (x2, y2) is given by rise m run the change in y the change in x y2 y1 y1 y2 x2 x1 x1 x2 Example Graph the function f (x) 2 x 1 and determine its slope. 3 Calculate two ordered pairs, plot the points, graph the function, and determine its slope. 2 f (3) (3) 1 2 1 1 3 2 f (9) 9 1 6 1 5 3 y2 y1 m x2 x1 5 1 4 2 93 6 3 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 0 2 2 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. 5 b. y 2 Choose any number for x ; y x y must be 5/2. 2 5/2 0 5/2 2 5/2 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? 1 m 12 m 0.083 8.3% 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%). Change in y Slope Average rate of change Change in x 55 6 49 1 6 2007 1999 8 8 The average rate of change over the 8-yr period was 1 an increase of 6 % per year. 8 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: 3x 6y 7 0 6y 3x 7 1 1 (6y) (3x 7) 6 6 1 7 y x 2 6 1 7 Thus, the slope is and the y-intercept is 0, . 2 6 Example 2 Graph y x 4 3 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. rise change in y 2 move 2 units down m run change in x 3 move 3 units right 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 M x 2.89x 70.64 The height, in centimeters, of an adult female with a humerus of length x is given by the function . F x 2.75x 71.48 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: f 26 2.75 26 71.48 142.98 Thus, the female was 142.98 cm tall.