# Transformations of Trigonometric Functions

The transformation 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

Example: Sketch the function $$y=\sin(x), y=\sin(x)+4,$$ and $$y=\sin(x+\frac{\pi}{2})$$.

Solution:

Here, $$y=\sin(x)$$ is shown in black. The curve $$y=\sin(x)+4$$ is shifted 4 units up, shown in green. The curve $$y=\sin(x+\frac{\pi}{2})$$ is shifted $$\frac{\pi}{2}$$ units to the left, and is shown in blue. Notice that this is the same thing as the curve $$y=\cos(x)$$.

## 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 $$c$$

$$y = \frac{1}{c} f(x)$$: 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 $$c$$

$$y = f(\frac{x}{c})$$: stretch the graph of $$y = f(x)$$ horizontally by a factor of $$c$$

Example: Sketch the functions $$y=\cos(x), y=\cos(2x),$$ and $$y=3\cos(2x)$$.

Solution:

The function$$y=\cos(x)$$ is shown in black. The blue curve represents the function $$y=\cos(2x)$$ and is compressed horizontally by a factor of 2. The green curve represents the function $$y=3\cos(2x)$$ and is the blue curve stretched vertically by a factor of 3.

## 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

Example: Given that the blue curve represents the function $$y=\sin(x)$$, determine whether the following statement is TRUE or FALSE.

The green curve on the following graph represents both $$y=-\sin(x)$$ and $$y=\sin(-x)$$.

Solution: TRUE - The green curve does represent $$y=-\sin(x)$$ and $$y=\sin(-x)$$. In this particular situation the reflection about the x-axis is the same as the reflection about the y-axis.

Example 1:

Example 2: