Determine vertical displacement of a sinusoidal function

Vertical displacement is the constant $d$ in a sinusoidal model such as $y=a\sin(b(x-h))+d$ or $y=a\cos(b(x-h))+d$ , shifting the graph upward when $d>0$ and downward when $d<0$ . It identifies the midline $y=d$ , which is the average of the maximum and minimum values; it must not be confused with amplitude, which measures the distance from the midline. The focus is on interpreting and determining this shift in standard sinusoidal equations, graphs, and applied models.

Detailed Explanation: 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.

Learn by doing: Determine vertical displacement of a sinusoidal function

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Sinusoidal Function Parameters (2 Params) - Graph to Function

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