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More on Functions and Graphs

Chapter 8. More on Functions and Graphs. Graphing and Writing Linear Functions. § 8.1. 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.

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More on Functions and Graphs

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  1. Chapter 8 More on Functions and Graphs

  2. Graphing and Writing Linear Functions § 8.1

  3. 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.

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

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

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

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

  8. 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).

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

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

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