More on Functions and Graphs Chapter 8 More on Functions and Graphs

Graphing and Writing Linear Functions § 8.1 Graphing and Writing Linear Functions

Linear Functions Identifying Linear Functions By the vertical line test, we know that all linear equations except those whose graphs are vertical lines are functions. Thus, all linear equations except those of the form x = c (vertical lines) are linear functions.

Graphing Linear Functions Example: Graph the linear function f (x) = x + 3. Let x = 4. f (4) = (4) + 3 Replace x with 4. f (4) = 3 + 3 = 6 Simplify the right side. One solution is (4, 6). Continued.

Graphing Linear Functions Example continued: Graph the linear function f (x) = x + 3. For the second solution, let x = 0. f (0) = (0) + 3 Replace x with 0. f (0) = 0 + 3 = 3 Simplify the right side. So a second solution is (0, 3). Continued.

Graphing Linear Functions Example continued: Graph the linear function f (x) = x + 3. For the third solution, let x = – 4. f (– 4) = (– 4) + 3 Replace x with – 4. f (– 4) = – 3 + 3 = 0 Simplify the right side. The third solution is (– 4, 0). Continued.

Graphing Linear Functions x y Example continued: (4, 6) (0, 3) (– 4, 0) Plot all three of the solutions (4, 6), (0, 3) and (– 4, 0). Draw the line that contains the three points.

Writing Linear Functions Example: Find an equation of the line whose slope is 5 and contains the point (4, 3). Write the equation using function notation. m = 5, x1 = 4, y1 = 3 y – y1 = m(x – x1) y – (– 3) = 5(x – 4) Substitute the values for m, x1, and y1. y + 3 = 5x – 20 Simplify and distribute. y = 5x – 23 Subtract 3 from both sides. f (x) = 5x – 23 Replace y with f (x).

Writing Linear Functions Example: Write a function that describes the line containing the point (4, 1) and is perpendicular to the line 5x – y = 20  y =  5x + 20 Solve the equation for y to find the slope from the slope-intercept form. y = 5x  20 5 is the slope of the line perpendicular to the one needed. As perpendicular lines have slopes that are negative reciprocals of each other, the slope of the line we want is Continued. 9

Writing Linear Functions Example: Write a function that describes the line containing the point (4, 1) and is perpendicular to the line 5x – y = 20 m = , x1 = 4, y1 = 1 y – y1 = m(x – x1) y – (1) = (x – 4) Substitute the values for m, x1, and y1. y + 1 = x Simplify and distribute. y = x Subtract 1 from both sides. f (x) = x Replace y with f (x). 10