Skill: Apply transformations to trigonometric functions

A transformed trigonometric function can be written as

$$y=a\sin(b(x-h))+k,\qquad y=a\cos(b(x-h))+k,$$

or

$$y=a\tan(b(x-h))+k.$$

The parameters have these meanings:

• $a$: vertical stretch or compression; if $a<0$, reflect across the $x$-axis.
• $b$: horizontal scale factor. The period is
  • $\dfrac{2\pi}{ \vert b \vert }$ for sine and cosine,
  • $\dfrac{\pi}{ \vert b \vert }$ for tangent.
• $h$: horizontal shift, right if $h>0$ and left if $h<0$.
• $k$: vertical shift, up if $k>0$ and down if $k<0$.

Example

Analyze and sketch

$$y=-2\cos\left(3\left(x-\frac{\pi}{6}\right)\right)+1.$$

Step 1: Identify the parameters

Compare the equation with

$$y=a\cos(b(x-h))+k.$$

Therefore,

$$a=-2,\qquad b=3,\qquad h=\frac{\pi}{6},\qquad k=1.$$

This tells us:

• The graph is reflected across the $x$-axis because $a$ is negative.
• It has a vertical stretch by a factor of $2$.
• It is horizontally compressed because $b=3$.
• It shifts right by $\dfrac{\pi}{6}$.
• It shifts up by $1$.

Step 2: Find the period

For cosine,

$$\text{Period}=\frac{2\pi}{ \vert b \vert }.$$

So,

$$\text{Period}=\frac{2\pi}{3}.$$

A cosine cycle is divided into four equal sections, so the distance between key points is

$$\frac{\text{Period}}{4} =\frac{2\pi/3}{4} =\frac{\pi}{6}.$$

Step 3: Find the key points

The first key point occurs at the phase shift:

$$x=h=\frac{\pi}{6}.$$

Starting with the standard cosine values, use the five points across one cycle:

$$1,\quad 0,\quad -1,\quad 0,\quad 1.$$

Substitute these into the transformed function $y=-2\cos(\cdots)+1$:

$$\begin{array}{c|c|c} x & \cos(\cdots) & y=-2\cos(\cdots)+1\\ \hline \frac{\pi}{6} & 1 & -1\\ \frac{\pi}{3} & 0 & 1\\ \frac{\pi}{2} & -1 & 3\\ \frac{2\pi}{3} & 0 & 1\\ \frac{5\pi}{6} & 1 & -1 \end{array}$$

The key points are

$$\left(\frac{\pi}{6},-1\right),\ \left(\frac{\pi}{3},1\right),\ \left(\frac{\pi}{2},3\right),\ \left(\frac{2\pi}{3},1\right),\ \left(\frac{5\pi}{6},-1\right).$$

Plot these points and connect them with a smooth cosine curve. The pattern repeats every $\dfrac{2\pi}{3}$.

Step 4: State the range and domain

The midline is

$$y=k=1.$$

The amplitude is

$$\vert a \vert =2.$$

Therefore, the maximum value is

$$1+2=3,$$

and the minimum value is

$$1-2=-1.$$

So the range is

$$-1\le y\le 3.$$

The domain of a sine or cosine function is all real numbers:

$$x\in\mathbb{R}.$$

Thus, the transformed graph has period $\dfrac{2\pi}{3}$, range $[-1,3]$, and domain $\mathbb{R}$.