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Any sinusoidal function written as A·sin(Bx+C)+D (or with cosine in place of sine) has four key parameters that fully describe its graph: amplitude (how tall the wave rises and falls above and below its midline), period (how long, in x-units, one complete cycle takes), phase shift (how far the wave is shifted horizontally left or right compared to a plain sin(x) or cos(x)), and vertical shift (how far the midline itself sits above or below the x-axis). This decomposition is exactly how engineers describe an AC voltage waveform, how physicists describe a swinging pendulum's position over time, and how audio engineers describe a sound wave's pitch and loudness — reading off these four numbers turns an intimidating-looking equation into an intuitive physical picture.
How it works
Enter the coefficients A, B, C, and D from your function A·sin(Bx+C)+D (or the cosine equivalent). The calculator computes amplitude = |A| (always taken as positive, since amplitude describes a distance), period = 2π/|B| (a larger B compresses the wave into a shorter period), phase shift = −C/B (found by factoring Bx+C as B(x + C/B), which reveals the horizontal shift is −C/B, not simply −C), and vertical shift = D (the height of the wave's new midline).
- Enter function.
- Enter amplitude (A).
- Enter b coefficient (of x).
- Enter c (phase term).
- Enter d (vertical shift).
- Click Calculate to see your results.
Examples
y = 2sin(3x + π/2) + 1
Amplitude = |2| = 2, period = 2π/3 ≈ 2.094, phase shift = −(π/2)/3 = −π/6 ≈ −0.524 (shifted left), vertical shift = 1.
y = −3cos(2x − π) − 4
Amplitude = |−3| = 3, period = 2π/2 = π ≈ 3.1416, phase shift = −(−π)/2 = π/2 ≈ 1.571 (shifted right), vertical shift = −4.
y = 0.5sin(4x) + 2
With C = 0: amplitude = 0.5, period = 2π/4 = π/2 ≈ 1.5708, phase shift = 0 (no horizontal shift), vertical shift = 2.
Who should use it
- Analyzing an AC voltage or current waveform's properties.
- Describing a pendulum, spring, or tidal cycle's motion over time.
- Precalculus and trigonometry coursework on graphing transformations.
Industry applications
- Electrical engineering (AC signal analysis)
- Physics (oscillatory motion)
- Audio engineering (waveform analysis)
Advantages
- Extracts all four defining parameters from one equation in a single step.
- Works for both sine- and cosine-based functions.
Limitations
- Assumes the equation is already in the standard A·sin(Bx+C)+D form — a function not yet in that form needs algebraic rearrangement first.
Common mistakes to avoid
- Forgetting to divide C by B before interpreting the phase shift — the shift is −C/B, not −C.
- Reporting a negative amplitude when A is negative, instead of taking its absolute value and treating the sign as a vertical reflection.
- Confusing phase shift (horizontal movement) with vertical shift (vertical movement) — they come from different coefficients (C versus D) entirely.
Best practices
- Always factor out B from inside the parentheses before reading off the phase shift, to avoid the −C/B versus −C mistake.
- Sketch or visualize the midline (y = D) first, then layer the amplitude, period, and phase shift on top of it for a clearer mental picture.
- When B is negative, consider rewriting the function with a positive B using the sine or cosine odd/even identities for clarity.
Tips
- Need the (x, y) unit-circle coordinates for a specific angle instead? Use the Unit Circle Calculator.