Skill: Graph sine, cosine, and tangent functions

A sine or cosine graph can be built from the unit-circle values

$$0,\ 1,\ 0,\ -1,\ 0$$

over one cycle. Transformations change the graph’s height, width, and position.

For

$$y=a\sin(bx+c)+d \quad\text{or}\quad y=a\cos(bx+c)+d,$$

• Amplitude: $\vert a \vert$
• Period: $\dfrac{2\pi}{ \vert b \vert }$
• Phase shift: $\displaystyle -\frac{c}{b}$
• Vertical translation: $d$

The midline is $y=d$. If $a<0$, the graph is reflected across its midline.

Worked example

Graph

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

1. Identify the transformations

Compare the equation with

$$y=a\sin(bx+c)+d.$$

Here,

$$a=2,\qquad b=1,\qquad c=-\frac{\pi}{3},\qquad d=1.$$

Therefore:

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

$$\frac{2\pi}{1}=2\pi$$

• Phase shift: $\displaystyle \frac{\pi}{3}$ units to the right
• Midline: $y=1$

The maximum value is

$$1+2=3,$$

and the minimum value is

$$1-2=-1.$$

2. Divide one period into four equal parts

A sine graph uses five key points over one period:

$$0,\quad \frac{\pi}{2},\quad \pi,\quad \frac{3\pi}{2},\quad 2\pi.$$

Since the graph is shifted right by $\frac{\pi}{3}$, add $\frac{\pi}{3}$ to each $x$-coordinate:

$$\begin{array}{c|ccccc} x & \frac{\pi}{3} & \frac{5\pi}{6} & \frac{4\pi}{3} & \frac{11\pi}{6} & \frac{7\pi}{3}\\ \hline y & 1 & 3 & 1 & -1 & 1 \end{array}$$

The $y$-values come from the usual sine pattern, stretched by $2$ and shifted up $1$.

3. Plot and connect the points

Plot

$$\left(\frac{\pi}{3},1\right),\quad \left(\frac{5\pi}{6},3\right),\quad \left(\frac{4\pi}{3},1\right),\quad \left(\frac{11\pi}{6},-1\right),\quad \left(\frac{7\pi}{3},1\right).$$

Join them with a smooth sine curve. Then repeat the same pattern every $2\pi$ units.

The graph has:

• Midline $y=1$
• Maximum $y=3$
• Minimum $y=-1$
• Period $2\pi$

For a cosine graph, use the same process, but the basic key points begin at a maximum:

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

For tangent,

$$y=a\tan(bx+c)+d,$$

there is no amplitude. Its period is

$$\frac{\pi}{|b|},$$

and its vertical asymptotes occur where

$$bx+c=\frac{\pi}{2}+k\pi,\qquad k\in\mathbb Z.$$

Between the asymptotes, tangent branches pass through the midline $y=d$ and repeat every period.