Sullivan PreCalculus Section 2.5 Graphing Techniques: Transformations

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Presentation transcript:

Sullivan PreCalculus Section 2.5 Graphing Techniques: Transformations 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

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

(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 f shifted vertically up (for c > 0).

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

(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 f shifted vertically down (for c > 0).

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

(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 f shifted horizontally right (if c > 0) or left (if c < 0).

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

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

(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).

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).

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

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

(-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).

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

(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).

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