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Chapter 8 More on Functions and Graphs

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§ 8.1 Graphing and Writing Linear Functions

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 33 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. Linear Functions

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 44 Let x = 4. f (4) = 3 + 3 = 6 S implify the right side. One solution is (4, 6). Graph the linear function f (x) = x + 3. f (4) = (4) + 3 R eplace x with 4. Graphing Linear Functions Example: Continued.

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 55 For the second solution, let x = 0. f (0) = 0 + 3 = 3 S implify the right side. So a second solution is (0, 3). Graph the linear function f (x) = x + 3. f (0) = (0) + 3 R eplace x with 0. Graphing Linear Functions Example continued: Continued.

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 66 For the third solution, let x = – 4. f (– 4) = – 3 + 3 = 0 S implify the right side. The third solution is ( – 4, 0). Graph the linear function f (x) = x + 3. f ( – 4) = ( – 4) + 3 Replace x with – 4. Example continued: Graphing Linear Functions Continued.

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 77 Plot all three of the solutions (4, 6), (0, 3) and (– 4, 0). x y (4, 6) (0, 3) (– 4, 0) Draw the line that contains the three points. Graphing Linear Functions Example continued:

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 88 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, x 1 = 4, y 1 = 3 Writing Linear Functions y – y 1 = m(x – x 1 ) y – (– 3) = 5(x – 4) y + 3 = 5x – 20 y = 5x – 23 Substitute the values for m, x 1, and y 1. Simplify and distribute. Subtract 3 from both sides. f (x) = 5x – 23 Replace y with f (x).

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 99 Example: Write a function that describes the line containing the point (4, 1) and is perpendicular to the line y = 5x + 20 Writing Linear Functions Solve the equation for y to find the slope from the slope-intercept form. 5 is the slope of the line perpendicular to the one needed. 5x – y = 20 y = 5x 20 As perpendicular lines have slopes that are negative reciprocals of each other, the slope of the line we want is Continued.

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

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