Graphing Sine and Cosine: Amplitude, Period and Phase Shift

Every sine or cosine graph can be written as $y = A\sin(Bx + C) + D$. The four numbers control the shape: $|A|$ is the amplitude, $\frac{2\pi}{|B|}$ is the period, $-\frac{C}{B}$ is the phase shift, and $D$ moves the midline up or down.

The basic wave

$y = \sin x$ starts at 0, rises to 1 at $\frac{\pi}{2}$, returns to 0 at $\pi$, drops to $-1$ at $\frac{3\pi}{2}$ and repeats every $2\pi$. The cosine graph is the same wave shifted left by $\frac{\pi}{2}$.

What each parameter does

Worked example 1: $y = 3\sin(2x) + 1$

• Amplitude $3$.
• Period $\frac{2\pi}{2} = \pi$.
• Midline $y = 1$, so the wave runs from $1 - 3 = -2$ to $1 + 3 = 4$.

Worked example 2: $y = 2\cos\left(x - \frac{\pi}{4}\right)$

Here $C = -\frac{\pi}{4}$, so the phase shift is $-\frac{C}{B} = \frac{\pi}{4}$: the graph moves right by $\frac{\pi}{4}$. The amplitude is 2 and the period is $2\pi$.

Negative A

A negative $A$ flips the wave upside down. $y = -\sin\frac{x}{2} - 2$ has amplitude 1, period $4\pi$, and midline $y = -2$.

Common mistakes

• Taking the period to be $2\pi B$ instead of $\frac{2\pi}{B}$.
• Reading the phase shift as $C$ instead of $-\frac{C}{B}$.
• Forgetting that amplitude is never negative.

Practice questions

1. Period of $y = \sin(3x)$?
2. Range of $y = 4\cos x - 1$?
3. Phase shift of $y = \sin(2x + \pi)$?

Answers: 1) $\frac{2\pi}{3}$ 2) $[-5, 3]$ 3) $-\frac{\pi}{2}$ (left by $\frac{\pi}{2}$)

The sine graph calculator draws two full periods with the midline marked. For individual values use the trigonometry calculator, and revise the basics in sine, cosine and tangent explained.

Frequently asked questions

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