Shape of the basic graphs
The graph of y = sin x oscillates smoothly between −1 and 1, repeating every 360°. The graph of y = cos x has the same shape but starts at its maximum. The graph of y = tan x rises without bound and repeats every 180°, with vertical asymptotes where cos x = 0.
Transformations
For y = a sin bx + c, the number a controls the amplitude |a| (half the distance between maximum and minimum), b controls the period, and c shifts the graph vertically. Maximum = c + |a|, minimum = c − |a|.
To sketch any of these curves, first mark the amplitude to fix the height of the band the curve lives in, then divide one period into four equal parts to locate the maximum, the two zeros and the minimum in turn. Adding a constant c simply slides the whole band up or down without changing its shape or its period. Counting how often the curve crosses a horizontal line is the fastest way to count the solutions of an equation.
Key formula /
Amplitude = |a|. Period of sin/cos = 360°/b (or 2π/b); period of tan = 180°/b. Number of cycles in 0° to 360° equals b for sin/cos.
Worked example
For y = 3 sin 2x + 1, the amplitude is |3| = 3, the period is 360°/2 = 180°, the maximum is 1 + 3 = 4 and the minimum is 1 − 3 = −2. In one full revolution (0° to 360°) the curve completes 2 cycles.
Remember
- Larger b squeezes the graph (shorter period).
- Tangent period is 180°/b, not 360°/b.
- Count intersections with a horizontal line to count equation solutions.