CN College Algebra Ch. 2 Functions and Their Graphs 2.5: Graphing Techniques: Transformations Goals: Graph functions using horizontal and vertical shifts.

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Presentation transcript:

CN College Algebra Ch. 2 Functions and Their Graphs 2.5: Graphing Techniques: Transformations Goals: Graph functions using horizontal and vertical shifts. Graph functions using reflections about the x-axis or y-axis. Graph functions using compressions and stretches.

Transformations: Transformation: A transformation is a one-to- one correspondence between two sets of points. We’ll study reflections, scale changes, and translations.

Translations: Vertical Shift: If a real number k is added to the right side of a function y = f(x), the graph of the new function y = f(x) + k is the graph of f shifted vertically up (if k > 0) or down (if k < 0). Horizontal Shift: If the argument x of a function f is replaced by x – h, h a real number, the graph of the new function y = f(x – h) is the graph of f shifted horizontally left (if h 0).

Reflections: Reflection about the x-axis: 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). Reflection about the y-axis: When the graph of the function y = f(x) is known, the graph of the new function y = f(-x) is the reflection about the y-axis of the graph of the function y = f(x).

Scale Changes: Vertical Scale Change: When the right side of a function y = f(x) is multiplied by a positive number a, the graph of the new function y = a f(x) is obtained by multiplying each y- coordinate on the graph of y = f(x) by a. The new graph is a vertical compression (if 0 1) of the graph of y = f(x).

Scale Changes: Horizontal Scale Change: If the argument x of a function y = f(x) is multiplied by a positive number a, the graph of the new function y = f(ax) is obtained by multiplying each x-coordinate of y = f(x) by 1/a. A horizontal compression results if a > 1, and a horizontal stretch occurs of 0 < a < 1.

Assignments: Homework: 2.5 Exercises #1-16, (odds), 63. Midterm: Wednesday, February 22.