Amplitude Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Write the equation of a cosine function with amplitude

4

, period

\pi

, midline

y = -1

, and a maximum at

x = 0

1
Amplitude $= 4$ , so $A = 4$ (positive since max is at $x = 0$ for cosine). Period $= \frac{2\pi}{B} = \pi$ , so $B = 2$ . Midline: $D = -1$ .
2
Since cosine has its maximum at $x = 0$ with no phase shift needed: $y = 4\cos(2x) - 1$ .

Answer

y = 4\cos(2x) - 1

Building a sinusoidal equation from its characteristics: amplitude gives

|A|

, the sign of

A

and the position of the maximum determine whether to use

\cos

\sin

(or add a phase shift), the period determines

B = 2\pi/\text{period}

, and the midline gives

D

About Amplitude

Amplitude is the maximum vertical distance from the midline of a periodic function to a peak or trough.

Find the amplitude of [formula].

Find the amplitude and midline of [formula].

A sinusoidal function has a maximum value of [formula] and a minimum value of [formula]. Find the am