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All courses Math III · F-TF.5 36 of 55
Choose trig functions to model periodic phenomena using amplitude, frequency, and midline.

Find the amplitude of a sinusoid from its maximum and minimum values

Problem
What is the amplitude of a sinusoid with these graph features: maximum 9, minimum 3?
Prompt sinusoidal graph on labeled axes showing only the equation or stated geometric source features; requested conclusions are withheld. Open full size
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Hint

Amplitude is half the distance between the maximum and minimum values.

Use (maximum - minimum)/2 because amplitude is half the vertical spread.

Solution walkthrough

01

Identify the extremes

\[\text{maximum}~=~9,~\text{minimum}~=~3\]

The question gives the top and bottom y-values of the sinusoid.

02

Find the vertical spread

\[9~-~3~=~6\]

Subtracting the minimum from the maximum gives the full vertical distance from bottom to top.

03

Take half the spread

\[6/2~=~3\]

Amplitude is the distance from the midline to a maximum or minimum, so it is half of the full spread.

04

State the amplitude

\[\text{amplitude}~=~3\]

The sinusoid's amplitude is 3 units.

+

Another way

  1. The midline is (9 + 3)/2 = 6, and the maximum 9 is 3 units above that midline.

!

Common mistake

Using 9 - 3 = 6 as the amplitude gives the full vertical spread. Amplitude is half of that spread.

Answer sinusoidal graph on labeled axes with the exact requested parameter, model, extremum, envelope, or comparison conclusion labeled.
amplitude = 3