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All courses Algebra II · F-TF.5 43 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: \(\text{maximum}~9\), \(\text{minimum}~3\)?
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

Use the extreme-value formula

\[\text{amplitude}=(\text{maximum}-\text{minimum})/2\]

Amplitude is half the total vertical distance from a sinusoid's minimum to its maximum.

02

Substitute the graph values

\[\text{amplitude}=(9-3)/2=6/2\]

The given maximum is 9 and the given minimum is 3, so their vertical separation is 6.

03

State the amplitude

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

The graph lies 3 units above and 3 units below its midline, confirming the half-range calculation.

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Another way

  1. First find the midline (9+3)/2=6, then compute the distance 9-6=3 from the midline to the maximum.

!

Common mistake

Do not use 9-3=6 as the amplitude; that is the full vertical range. Divide by 2.

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

Solution walkthrough video