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Transformations of Trigonometric Functions

The transpformation of functions includes the shifting, stretching, and reflecting of their graph. The same rules apply when transforming trigonometric functions.

Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph of

y = f(x) + c: Shift the graph of y = f(x) up by c units

y = f(x) - c: Shift the graph of y = f(x) down by c units

y = f(x - c): Shift the graph of y = f(x) to the right by c units

y = f(x + c): Shift the graph of y = f(x) to the left by c units

Trigonometric transformations

Vertical and Horizontal Stretches/Compressions

Suppose c > 1. To obtain the graph of

y = cf(x): stretch the graph of y = f(x) vertically by a factor of c

y = 1/c f(x): compress the graph of y = f(x) vertically by a factor of c

y = f(cx): compress the graph of y = f(x) horizontally by a factor of c

y = f(x/c): stretch the graph of y = f(x) horizontally by a factor of c

Trigonometric transformations

Reflections

To obtain the graph of 

y = -f(x): reflect the graph of y = f(x) about the x-axis; and

y = f(-x): reflect the graph of y = f(x) about the y-axis

Trigonometric transformations

Example 1:

Example 2:

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