Skill: Graph sine functions

To graph a sine function, rewrite it in the form

$$y=a\sin\bigl(b(x-h)\bigr)+d.$$

Identify these features:

• Amplitude: $\vert a \vert$, the distance from the midline to a maximum or minimum
• Period: $\dfrac{2\pi}{ \vert b \vert }$, the length of one complete cycle
• Horizontal shift: $h$
• Midline: $y=d$
• Range: from $d- \vert a \vert$ to $d+ \vert a \vert$

Then plot five key points for one cycle: the starting point, maximum, midline, minimum, and ending point.

Example

Graph

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

1. Identify the transformations

Compare the equation with

$$y=a\sin\bigl(b(x-h)\bigr)+d.$$

Here,

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

Therefore:

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

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

• Horizontal shift: right $\dfrac{\pi}{3}$
• Midline: $y=1$

The graph’s highest and lowest values are

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

So the range is

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

2. Divide the period into four equal parts

One period is $2\pi$, so each quarter-period is

$$\frac{2\pi}{4}=\frac{\pi}{2}.$$

The cycle begins at the horizontal shift, $x=\dfrac{\pi}{3}$. Add $\dfrac{\pi}{2}$ each time:

$$\begin{array}{c|c} x & y \\ \hline \dfrac{\pi}{3} & 1 \\ \dfrac{\pi}{3}+\dfrac{\pi}{2}=\dfrac{5\pi}{6} & 3 \\ \dfrac{\pi}{3}+\pi=\dfrac{4\pi}{3} & 1 \\ \dfrac{\pi}{3}+\dfrac{3\pi}{2}=\dfrac{11\pi}{6} & -1 \\ \dfrac{\pi}{3}+2\pi=\dfrac{7\pi}{3} & 1 \end{array}$$

Because the coefficient $a=2$ is positive, the graph starts on the midline and rises to its maximum.

3. Plot and connect the points

Plot the five points

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

Connect them with a smooth sine curve, then repeat the pattern in both directions.

The graph oscillates around the midline $y=1$, reaches a maximum of $3$, a minimum of $-1$, and completes one cycle from $x=\dfrac{\pi}{3}$ to $x=\dfrac{7\pi}{3}$.