The Four-Parameter Machine

Every transformation of sin x is controlled by four numbers. Change one at a time using the explorer below, then read the analysis underneath.

$y = A \cdot \sin(Bx + C) + D$

What each parameter does

A — amplitude. Vertical stretch. The curve oscillates between D − |A| and D + |A|. If A is negative, the curve flips about the midline — peaks become troughs and vice versa. The amplitude is always |A|, never negative.

B — frequency. Controls how compressed the wave is horizontally. Period = 2π / |B|. If B = 2, the wave completes a full cycle in π instead of 2π — it fits twice as many cycles into the same x-range. If B is negative, the curve reflects about the y-axis (for sin, this is equivalent to a sign change in A).

C — phase shift. The most commonly confused parameter. The shift is not C — it is −C/B. When C = π/2 and B = 1, the shift is −π/2, meaning the curve moves left by π/2. Test yourself: set B = 2, C = π/2 in the explorer. What is the shift? (Answer: −π/4.)

D — vertical shift (midline). Lifts the entire curve up or down. The midline moves from y = 0 to y = D. The orange dashed line in the explorer shows the midline whenever D ≠ 0.

Reading a graph backward

Given a graph, extract the four parameters in this order:

1. Read D from the midline (halfway between max and min)
2. Read A from the amplitude (half the total height)
3. Read the period, compute B = 2π / period
4. Read where the curve crosses the midline going upward — that x-value gives the phase shift, from which C = −B × shift

Worked example. A curve has maximum 5, minimum −1, period π, and crosses its midline going up at x = π/6.

• D = (5 + (−1)) / 2 = 2
• A = (5 − (−1)) / 2 = 3
• B = 2π / π = 2
• Phase shift = π/6, so C = −2 × π/6 = −π/3

Answer: y = 3 sin(2x − π/3) + 2.

The period trap

JEE often gives y = sin(2x + π/3) and asks for the period. Students write π/3. The period is 2π / |B| = 2π / 2 = π. C has nothing to do with the period. B alone controls it.

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Next: Modulus transformations — |f(x)| and f(|x|) are completely different, and JEE knows it.