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Sullivan Algebra and Trigonometry: Section 3.4

Learn how to graph functions using horizontal and vertical shifts, compressions and stretches, and reflections about the x-axis and y-axis. Examples and points on the graphs are provided.

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Sullivan Algebra and Trigonometry: Section 3.4

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  1. Sullivan Algebra and Trigonometry: Section 3.4 • Objectives • Graph Functions Using Horizontal and Vertical Shifts • Graph Functions Using Compressions and Stretches • Graph Functions Using Reflections about the x-Axis and y-Axis

  2. Example: Use the graph of to obtain the graph of First, find points on the graph of f and g.

  3. (2, 6) (1, 3) (2, 4) (0, 2) (1, 1) If a real number c is added to the right side of a function y = f(x), the graph of the new function y = f(x) + c is the graph of fshifted vertically up (for c> 0).

  4. Example: Use the graph of to obtain the graph of First, find points on the graph of f and g.

  5. (2, 4) (1, 1) (2, 1) (1, -2) (0, -3) If a real number c is subtracted from the right side of a function y = f(x), the graph of the new function y = f(x) - c is the graph of fshifted vertically down (for c> 0).

  6. Example: Use the graph of to obtain the graph of First, find points on the graph of f and g.

  7. (0, 4) (2, 4) (-1, 1) (-3, 1) (1, 1) (-2, 0) If a real number c is subtracted from the argument x of a function y = f(x), the graph of the new function y = f(x - c) is the graph of fshifted horizontally right (if c > 0) or left (if c < 0).

  8. (-1, 4) (2, 4) (0, 0) (-3, 0) (-1, 2) (-3, -2) Example: Graph the function

  9. Example: Use the graph of to obtain the graph of First, find points on the graph of f and g.

  10. (2, 4) (1, 2) (2, 2) (1, 1) When the right side of a function y = f(x) is multiplied by a positive number k, the graph of the new function y = kf (x) is a vertically compressed (if 0 < k < 1) vertically stretched (if k > 1) version of the graph of y = f(x).

  11. Example: Graph each of the following functions: Y2 Y1 Y3 When the argument of a function y = f (x) is multiplied by a positive number k, the graph of the new function y = f(kx) is a horizontally stretched (if 0 < k < 1) or horizontally compressed (if k > 1) version of the graph of y = f (x).

  12. (4, 2) (4, 6) (8, 6) Example: Graph

  13. Example: Use the graph of to obtain the graph of First, find points on the graph of f and g.

  14. (-3, 9) (2, 4) (2, -4) (-3, -9) When the right side of a function y = f(x) is multiplied by -1, the graph of the new function y = -f (x) is the reflection about the x-axis of the graph of the function y = f(x).

  15. Example: Use the graph of to obtain the graph of First, find points on the graph of f and g.

  16. (9, 3) (-9, 3) (-4, 2) (4, 2) (-1, 1) (1, 1) When the argument of a function y = f (x) is multiplied by -1, the graph of the new function y = f(-x) is the reflection about the y-axis of the graph of y = f (x).

  17. (-4,2) (4,2) (-1,1) (1,1) (-1,-1) (-4,-2) Example: Graph

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