1.3 Linear Functions, Slope, and Applications 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 yintercept. Solve applied problems involving linear functions. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 2 Examples Linear Function y = mx + b y 1 x 2 5 Identity Function y = 1•x + 0 or y = x Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 3 Examples Constant Function y = 0•x + b or y = -2 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Not a Function Vertical line: x = 4 Slide 1.3 - 4 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 5 Example Graph the function 2 x 3 y 3 and determine its slope. Solution: Calculate two ordered pairs, plot the points, graph the function, and determine its slope. x 3: 2(3) 3 y 3 3 y 3 6 3 y 1; x 9: (3, 1) 2(9) 3 y 3 3 y 3 18 15 y 5; (9, 5) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 6 (3,1) (9,5) y2 y1 m x2 x1 5 1 4 2 93 6 3 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 7 Types of Slopes Positive—line slants up from left to right Negative—line slants down from left to right Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 8 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 9 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 10 Example Graph each linear equation and determine its slope. a. x = –2 Choose any number for y ; x must be –2. x ‒2 ‒2 ‒2 y 3 0 ‒4 m y2 y1 x2 x1 30 3 2 (2) 0 Vertical line 2 units to the left of the y-axis. Slope is not defined. Not the graph of a function. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 11 Example (continued) Graph each linear equation and determine its slope. 5 b. y 2 5 Choose any number for x ; y must be . 2 x y y2 y1 m x2 x1 52 0 5 5 –3 5 2 0 2 2 0 52 1 3 0 3 Horizontal line 5/2 units above the x-axis. Slope 0. The graph is that of a constant function. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 12 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/falls 4 ft for every horizontal distance of 100 ft. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 13 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%. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 14 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 15 Example The percent of American adolescents ages 12 to 19 who are obese increased from about 6.5% in 1985 to 18% in 2008. The graph below illustrates this trend. Find the average rate of change in the percent of adolescents who are obese from 1985 to 2008. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 16 Example The coordinates of the two points on the graph are (1985, 6.5%) and (2008, 18%). Change in y Slope Average rate of change Change in x 18% 6.5% 11.5% 0.5% 2008 1985 23 The average rate of change over the 23-yr period was an increase of 0.5% per year. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 17 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). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 18 Example Find the slope and y-intercept of the line with equation y = –0.25x – 3.8. Solution: y = –0.25x – 3.8 Slope = –0.25; y-intercept = (0, –3.8) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 19 Example Find the slope and y-intercept of the line with equation 3x – 6y 7 = 0. Solution: 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 20 Example 2 Graph y x 4 3 Solution: 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 21 Example There is no proven way to predict a child’s adult height, but there is a linear function that can be used to estimate the adult height of a child, given the sum of the child’s parents heights. The adult height M, in inches of a male child whose parents’ total height is x, in inches, can be estimated with the function M x 0.5x 2.5. The adult height F, in inches, of a female child whose parents’ total height is x, in inches, can be estimated with the function F x 0.5x 2.5. Estimate the height of a female child whose parents’ total height is 135 in. What is the domain of this function? Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 22 Example Solution: We substitute into the function: F x 0.5x 2.5. F 135 0.5 135 2.5 65 Thus we can estimate the adult height of the female child as 65 in., or 5 ft 5 in. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.3 - 23

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