Graph cosine functions

The learner interprets and graphs cosine functions in forms such as $y=a\cos(b(x-h))+k$ , relating the parameters to amplitude $|a|$ , period $2\pi/|b|$ , horizontal shift $h$ , reflection when $a<0$ , and vertical shift $k$ . They identify the midline, maximum and minimum values, range, key points, and periodic extension in radians, while distinguishing horizontal scaling from vertical scaling; inverse, complex-valued, and calculus-based analyses are not included.

Detailed Explanation: 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.

Learn by doing: Graph cosine functions

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