You know how to stretch and squeeze graphs: multiplying the output by $a$ stretches vertically, and replacing $x$ with $bx$ squeezes horizontally by a factor of $\frac{1}{b}$. Apply those moves to the sine wave and you control how tall the wave is and how long one cycle lasts.
Idea clave
For $y = A\sin(Bx)$ or $y = A\cos(Bx)$ with $B > 0$: the amplitude is $|A|$ and the period is $\dfrac{2\pi}{B}$.
Por qué funciona
Amplitude. Every height of $\sin(Bx)$ lies between $-1$ and $1$; multiplying by $A$ makes it lie between $-|A|$ and $|A|$. A negative $A$ also flips the wave upside down.
Period. One full cycle of sine needs its input to run from $0$ to $2\pi$. The input here is $Bx$, and $Bx$ goes from $0$ to $2\pi$ while $x$ goes from $0$ to $\frac{2\pi}{B}$. So a bigger $B$ packs cycles closer together and makes the period shorter.
Period. One full cycle of sine needs its input to run from $0$ to $2\pi$. The input here is $Bx$, and $Bx$ goes from $0$ to $2\pi$ while $x$ goes from $0$ to $\frac{2\pi}{B}$. So a bigger $B$ packs cycles closer together and makes the period shorter.
Worked examples
Ejemplo resuelto: Reading the numbers
Find the amplitude and period of $y = 3\sin(4x)$.
- Amplitude $= |3| = 3$.
- Period $= \dfrac{2\pi}{4} = \dfrac{\pi}{2}$.
- So the wave rises to $3$ and repeats four times as often as $\sin x$.
Respuesta: Amplitude $3$, period $\frac{\pi}{2}$
Ejemplo resuelto: A slow, flipped wave
Find the amplitude and period of $y = -2\cos\left(\frac{1}{2}x\right)$.
- Amplitude $= |-2| = 2$. The minus sign flips the wave: it starts at its lowest point, $-2$.
- Period $= \dfrac{2\pi}{1/2} = 4\pi$, twice as long as usual.
Respuesta: Amplitude $2$, period $4\pi$
Ejemplo resuelto: Working backward
Write $y = A\sin(Bx)$ with $A, B > 0$, amplitude $5$ and period $6$.
- $A = 5$.
- Solve $\dfrac{2\pi}{B} = 6$: $B = \dfrac{2\pi}{6} = \dfrac{\pi}{3}$.
Respuesta: $y = 5\sin\left(\frac{\pi}{3}x\right)$
Error común
Taking $B$ itself as the period, so that $y = \sin 4x$ would have period $4$. Bigger $B$ means more cycles in the same space, so the period must get smaller. Check: when $x$ equals one period, the input $Bx$ should equal exactly $2\pi$. For $\sin 4x$, $4 \cdot \frac{\pi}{2} = 2\pi$. ✓