Geometry Lab formula guide
Transforming sine & cosine graphs
For y = A sin(B(x − h)) + k, the amplitude is |A|, the period is 2π / |B|, the graph shifts h units horizontally, and the midline is y = k. Cosine graphs follow the same rules.
y = A sin(B(x − h)) + k
| Use it to | Formula |
|---|---|
| Amplitude | |A| |
| Period | 2π / |B| |
| Phase shift | h (right when positive) |
| Midline and range | y = k; range [k − |A|, k + |A|] |
Try it with your own numbers
Transforming sine & cosine graphs calculator
Four parameters shape an entire wave. Change a value and the diagram, result, and working update together.
Sine & cosine graph explorerLIVE EXPLORATION
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Your measurements
The graph spans −2π to 2π on the x-axis. x and h are measured in radians.
Why it works
y = A sin(B(x − h)) + k
The idea
For nonzero A and B, |A| is the amplitude, 2π/|B| is the period, h is the horizontal shift, and k is the vertical shift. Cosine follows the same rules. The midline is y = k and the output ranges from k − |A| to k + |A|.
Work through an example
1Work through an example
- For y = 2 sin(3(x − π/6)) + 1, the amplitude is 2, period is 2π/3, phase shift is π/6 to the right, and midline is y = 1. Its range is [−1, 3].
Common mistakes
- In y = A sin(Bx − C) + k, the phase shift is C/B, not C. If A = 0, the output is constant and there is no unique fundamental period.
Make the formula move.
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