 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 1.3 Linear Functions, Slope, and Applications

Linear Functions Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Linear Function y = mx + b Identity Function y = x

Horizontal and Vertical Lines Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Horizontal lines are given by equations of the type y = b or f(x) = b. They are functions. m = 0 Vertical lines are given by equations of the type x = a. They are not functions. m is undefined y =  2 x =  2

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

Types of Slopes Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Positive, m > 0 : line slants up from left to right Negative, m > 0 : line slants down from left to right

Example Graph each linear equation and determine its slope. a. x = –2 b. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Applications of Slope Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The grade, or slope, of the ramp is 8.3%. 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?

Average Rate of Change Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. Average rate of change formula: = Change in f(x) values change in x values

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The percent of American adolescents ages 12 to 19 who are obese increased from about 6.5% in 1985 to 18% in The graph below illustrates this trend. Find the average rate of change in the percent of adolescents who are obese from 1985 to 2008.

Slope-Intercept Equation Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The linear function f given by f (x) = mx + b is written in slope-intercept form. The constant m is called the slope, and the y-intercept is (0, b).

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Find the slope and y-intercept of the line with equation 3x – 6y  7 = 0.

Example – Using the Slope & Intercept Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graph using the slope and y-intercept

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley There is no proven way to predict a child’s height 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?