Amplitude, Period, and Midline of Sinusoids

Algebra Foundation

Algebra 2

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Algebra 2 · Unit 10 · Topic 10.4

Amplitude, Period, and Midline of Sinusoids

For y = A sin(B(x − h)) + k in radian measure with A and B nonzero, amplitude is |A|, period is 2π/|B|, and the midline is y = k.

Definition and explanation

For y = A sin(B(x − h)) + k in radian measure with A and B nonzero, amplitude is |A|, period is 2π/|B|, and the midline is y = k. The value h controls horizontal placement.

Worked example

For y = 3 sin(2x) − 1, amplitude is 3, period is π, and midline is −1. The range is [−4, 2]. At x = π/4, sin(π/2) = 1 gives the maximum output 2.

Common mistake

The maximum output is k + |A|, not simply the amplitude.

Key takeaway

For y = A sin(B(x − h)) + k in radian measure with A and B nonzero, amplitude is |A|, period is 2π/|B|, and the midline is y = k.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.