Skill: Determine vertical displacement of a sinusoidal function

In a sinusoidal equation written as

$$y=a\sin(b(x-h))+d \quad \text{or} \quad y=a\cos(b(x-h))+d,$$

the vertical displacement is the constant $d$ added outside the sine or cosine function.

• The midline is $y=d$.
• If $d>0$, the graph shifts upward.
• If $d<0$, the graph shifts downward.
• The amplitude is $\vert a \vert$, so do not confuse $a$ with $d$.

Example

Determine the vertical displacement and midline of

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

Step 1: Identify the constant outside the sine function.

The equation has the form

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

Comparing the equation with this form:

• $a=-3$
• $b=2$
• $h=\frac{\pi}{4}$
• $d=2$

Step 2: State the vertical displacement.

Since $d=2$, the graph is shifted up 2 units.

Therefore, the vertical displacement is

$$\boxed{2}.$$

Step 3: Identify the midline.

The midline is $y=d$, so the midline is

$$\boxed{y=2}.$$

The number $-3$ is not the vertical displacement. Its absolute value, $\vert -3 \vert =3$, is the amplitude. The amplitude tells us the distance from the midline, while $2$ tells us where the midline is located.