Trigonometry · step 4 of 13

Graphs of sin, cos & tan

For y = a sin(bx + c) + d, the amplitude is |a|, the period is 2π/b, the vertical shift is d and the horizontal (phase) shift is −c/b. That last one is the trap: the shift is −c/b, not −c, because b compresses the horizontal axis first. Tangent behaves differently — period π, no amplitude, and asymptotes where cosine vanishes.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Read the amplitude and vertical shift|a| is the half-range; d is the midline. The graph oscillates between d − |a| and d + |a|.
Compute the period as 2π/bLarger b means faster oscillation. For tangent the period is π/b.
Factor to find the phase shiftWrite bx + c as b(x + c/b); the shift is −c/b, to the left when positive.
Plot five key points per cycleStart, quarter, half, three-quarter, end — enough to draw one full period accurately.

Worked example

For y = 3 sin(2x − π/2) + 1, state the amplitude, period, phase shift and range.

Amplitude |a| = 3; vertical shift d = 1.
Period = 2π/b = 2π/2 = π.
Factor: 2x − π/2 = 2(x − π/4), so the phase shift is π/4 to the RIGHT.
Range: 1 ± 3.

Answer. Amplitude 3, period π, phase shift π/4 right, midline y = 1 and range [−2, 4].

Where marks get dropped

These are the specific errors that cost credit on graphs of sin, cos & tan questions — QED's rubric penalises each of them separately.

Reading the phase shift as −c. It is −c/b, so you must factor out b first.
Assuming tangent has an amplitude. It is unbounded, so amplitude is undefined; its period is π, not 2π.
Confusing the period with the frequency. Period = 2π/b, so a bigger b gives a SHORTER period.

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Graphs of sin, cos & tan — frequently asked questions

How are sine and cosine related as graphs?

cos x = sin(x + π/2) — cosine is sine shifted a quarter period left. Any sine graph can be written as a cosine and vice versa.

Where are the asymptotes of tan x?

At x = π/2 + kπ, where cos x = 0 and the ratio sin/cos is undefined. Between consecutive asymptotes tan increases from −∞ to ∞.

What does a negative amplitude mean?

A reflection in the midline. The amplitude itself is |a|, and the sign only flips the graph vertically.

The rest of Trigonometry

Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.

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