Amplitude and period

We compare the graph of $y = 2\sin 2x$ with the basic $y = \sin x$. In general $y = a\sin bx$ with $b > 0$ has amplitude $|a|$ and period $\dfrac{2\pi}{b}$.

Amplitude and period

Here $a = 2$ and $b = 2$, so the amplitude is $2$ and the period is $\dfrac{2\pi}{2} = \pi$.

The graph is stretched to twice the height and compressed to half the width.

The role of $a$

The role of $a$ is simple. The sine runs between $-1$ and $1$, so multiplying by $a$ makes it run between $-|a|$ and $|a|$. The graph is stretched vertically by a factor of $a$, and that is all.

The role of $b$

The role of $b$ takes a moment's thought. The sine completes one cycle when its argument increases by $2\pi$. The argument is $bx$, so writing $T$ for the increase in $x$ needed for one cycle gives the following.

The larger $b$ is, the faster the argument grows, so the wave is squeezed horizontally and the period shortens. With $b = 2$ a cycle completes in half the original span of $x$.

Zeros and maxima

The zeros and maxima crowd in by the same factor. The zeros of $y = \sin x$ come every $\pi$, while those of $y = 2\sin 2x$ come every $\dfrac{\pi}{2}$. The maximum occurs where the argument is $\dfrac{\pi}{2}$, so $2x = \dfrac{\pi}{2}$ gives $x = \dfrac{\pi}{4}$, and the value there is $2$.

The more general form

• $a$ is the amplitude, the vertical stretch
• $b$ fixes the period, the horizontal stretch
• $c$ is the horizontal shift, the phase
• $d$ is the vertical translation

Applications

For a sound wave, $a$ is the loudness and $b$ the pitch, the frequency being $\dfrac{b}{2\pi}$; for an alternating current, $a$ fixes the size of the voltage and $b$ the frequency. Amplitude and period are the first two things to look at whenever a wave is under discussion.

The large dots are the maximum of $2\sin 2x$ at $\left( \dfrac{\pi}{4}, 2 \right)$ and the maximum of $\sin x$ at $\left( \dfrac{\pi}{2}, 1 \right)$.