Chapter 1 Graphs Functions and Models Copyright 2016

Chapter 1 Graphs, Functions, and Models Copyright © 2016, 2012 Pearson Education, Inc. 1 -1

Section 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 Introduction to Graphing Functions and Graphs Linear Functions, Slope, and Applications Equations of Lines and Modeling Linear Equations, Functions, Zeros and Applications Solving Linear Inequalities Copyright © 2016, 2012 Pearson Education, Inc. 1 -2

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 © 2016, 2012 Pearson Education, Inc. 1 -3

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 © 2016, 2012 Pearson Education, Inc. 1 -4

Examples Linear Function y = mx + b Identity Function y = 1 • x + 0 or y = x Copyright © 2016, 2012 Pearson Education, Inc. 1 -5

Examples Constant Function y = 0 • x + b or y = b Not a Function Vertical line: x = a Copyright © 2016, 2012 Pearson Education, Inc. 1 -6

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. x= 2 y= 2 Copyright © 2016, 2012 Pearson Education, Inc. 1 -7

Slope The slope m of a line containing the points (x 1, y 1) and (x 2, y 2) is given by Copyright © 2016, 2012 Pearson Education, Inc. 1 -8

Example Graph the function and determine its slope. Solution: Calculate two ordered pairs, plot the points, graph the function, and determine its slope. Copyright © 2016, 2012 Pearson Education, Inc. 1 -9

Types of Slopes Positive—line slants up from left to right Negative—line slants down from left to right Copyright © 2016, 2012 Pearson Education, Inc. 1 -10

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 © 2016, 2012 Pearson Education, Inc. 1 -11

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 © 2016, 2012 Pearson Education, Inc. 1 -12

Example Graph each linear equation and determine its slope. a. x = – 2 Choose any number for y ; x must be – 2. x ‒ 2 ‒ 2 y 3 0 ‒ 4 Vertical line 2 units to the left of the y-axis. Slope is not defined. Not the graph of a function. Copyright © 2016, 2012 Pearson Education, Inc. 1 -13

Example (continued) Graph each linear equation and determine its slope. b. Choose any number for x ; y must be x y 0– 31 Horizontal line 5/2 units above the x-axis. Slope 0. The graph is that of a constant function. Copyright © 2016, 2012 Pearson Education, Inc. 1 -14

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. Copyright © 2016, 2012 Pearson Education, Inc. 1 -15

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%. Copyright © 2016, 2012 Pearson Education, Inc. 1 -16

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 © 2016, 2012 Pearson Education, Inc. 1 -17

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 © 2016, 2012 Pearson Education, Inc. 1 -18

Example The coordinates of the two points on the graph are (1985, 6. 5%) and (2008, 18%). The average rate of change over the 23 -yr period was an increase of 0. 5% per year. Copyright © 2016, 2012 Pearson Education, Inc. 1 -19

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 © 2016, 2012 Pearson Education, Inc. 1 -20

Example Find the slope and y-intercept of the line with equation y = – 0. 25 x – 3. 8. Solution: y = – 0. 25 x – 3. 8 Slope = – 0. 25; y-intercept = (0, – 3. 8) Copyright © 2016, 2012 Pearson Education, Inc. 1 -21

Example Find the slope and y-intercept of the line with equation 3 x – 6 y 7 = 0. Solution: We solve for y: Thus, the slope is and the y-intercept is Copyright © 2016, 2012 Pearson Education, Inc. 1 -22

Example Graph 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. Copyright © 2016, 2012 Pearson Education, Inc. 1 -23

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 The adult height F, in inches, of a female child whose parents’ total height is x, in inches, can be estimated with the function Estimate the height of a female child whose parents’ total height is 135 in. What is the domain of this function? Copyright © 2016, 2012 Pearson Education, Inc. 1 -24

Example Solution: We substitute into the function: Thus we can estimate the adult height of the female child as 65 in. , or 5 ft 5 in. Copyright © 2016, 2012 Pearson Education, Inc. 1 -25