Three numbers reshape the basic curves. In y = a·sin(bx) + c, a controls the amplitude (height), b controls the period (360°/b), and c shifts the whole graph vertically. Read them off, or reverse-engineer them from a graph's maximum and minimum.
The value |a| is the amplitude — half the vertical distance between the maximum and minimum. It stretches the curve up and down.
The value b squeezes the curve horizontally: the new period is 360° / b. So y = sin(2x) completes a full cycle in 180°.
amplitude a = (max − min) / 2 and vertical shift c = (max + min) / 2. These two formulas unlock most 'find a and c' questions.
State the amplitude and period of y = 3 sin(2x).
A curve y = a cos x + c has maximum 7 and minimum −1. Find a and c.
State the amplitude and period of y = 5 sin x.
State the period of y = sin(4x).
State the maximum and minimum values of y = 2 sin x + 3.
A curve y = a sin x + c has maximum 10 and minimum 2. Find a and c.
A curve of the form y = a cos(bx) has amplitude 3 and period 120°. Write down its equation.