# Transformations of Trigonometric Functions

### A QUICK REVIEW

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 functions  $$y = \sin(x)$$, $$y = \sin(x) +4$$, 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 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 $$c$$;

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