Skill: Graph cosine functions

To graph a cosine function in the form

$$y=a\cos\bigl(b(x-h)\bigr)+k,$$

identify these features:

• Amplitude: $\vert a \vert$
• Period: $\dfrac{2\pi}{ \vert b \vert }$
• Horizontal shift: $h$
• Vertical shift: $k$
• Midline: $y=k$
• Reflection: If $a<0$, the graph is reflected across its midline.
• Range: $[k- \vert a \vert ,\ k+ \vert a \vert ]$

The period is divided into four equal parts to find the five main points of one cycle.

Example

Graph

$$y=-2\cos\left(3\left(x-\frac{\pi}{4}\right)\right)+1.$$

Step 1: Identify the features

Compare the equation with

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

Thus,

$$a=-2,\qquad b=3,\qquad h=\frac{\pi}{4},\qquad k=1.$$

Therefore:

• Amplitude: $\vert a \vert =2$
• Period:

$$\frac{2\pi}{ \vert b \vert }=\frac{2\pi}{3}$$

• Horizontal shift: $\dfrac{\pi}{4}$ to the right
• Vertical shift: up $1$
• Midline: $y=1$
• Since $a<0$, the cosine graph is reflected.

The maximum and minimum values are

$$1+2=3 \qquad\text{and}\qquad 1-2=-1.$$

So the range is

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

Step 2: Divide the period into four parts

The quarter-period is

$$\frac{1}{4}\left(\frac{2\pi}{3}\right)=\frac{\pi}{6}.$$

Start at the horizontal shift $x=\dfrac{\pi}{4}$. Add $\dfrac{\pi}{6}$ each time:

$$\frac{\pi}{4},\quad \frac{5\pi}{12},\quad \frac{7\pi}{12},\quad \frac{3\pi}{4},\quad \frac{11\pi}{12}.$$

Step 3: Find the five key points

Because the graph is reflected, it starts at a minimum:

$x$	$y$	Reason
$\dfrac{\pi}{4}$	$-1$	minimum
$\dfrac{5\pi}{12}$	$1$	midline
$\dfrac{7\pi}{12}$	$3$	maximum
$\dfrac{3\pi}{4}$	$1$	midline
$\dfrac{11\pi}{12}$	$-1$	minimum

Plot these points and connect them with a smooth cosine curve. Draw the midline $y=1$ as a guide.

The pattern repeats every $\dfrac{2\pi}{3}$ units horizontally. For example, adding or subtracting $\dfrac{2\pi}{3}$ from every $x$-coordinate gives the next or previous cycle.