8.5 frequency and period of sinusoidal functions practice find the frequency and period of each function…

Explanation:

Step1: Recall the general form

The general form of a sinusoidal function is $y = A\sin(Bx - C)+D$ or $y=A\cos(Bx - C)+D$. The period $T$ is given by $T=\frac{2\pi}{|B|}$ and the frequency $f$ is given by $f = \frac{|B|}{2\pi}$.

Step2: For $f(x)=\sin(4x)+1$

Here $B = 4$. Period $T=\frac{2\pi}{|4|}=\frac{\pi}{2}$. Frequency $f=\frac{|4|}{2\pi}=\frac{2}{\pi}$.

Step3: For $g(x)=- 3\cos(2x)$

Here $B = 2$. Period $T=\frac{2\pi}{|2|}=\pi$. Frequency $f=\frac{|2|}{2\pi}=\frac{1}{\pi}$.

Step4: For $h(x)=\cos(\frac{1}{2}x)+2$

Here $B=\frac{1}{2}$. Period $T=\frac{2\pi}{\left|\frac{1}{2}\right|}=4\pi$. Frequency $f=\frac{\left|\frac{1}{2}\right|}{2\pi}=\frac{1}{4\pi}$.

Step5: For $k(x)=-2\sin(\frac{3}{4}x)+1$

Here $B = \frac{3}{4}$. Period $T=\frac{2\pi}{\left|\frac{3}{4}\right|}=\frac{8\pi}{3}$. Frequency $f=\frac{\left|\frac{3}{4}\right|}{2\pi}=\frac{3}{8\pi}$.

Step6: For $j(x)=4\cos(3x)-1$

Here $B = 3$. Period $T=\frac{2\pi}{|3|}=\frac{2\pi}{3}$. Frequency $f=\frac{|3|}{2\pi}=\frac{3}{2\pi}$.

Answer:

1. $f(x)=\sin(4x)+1$: Period $T = \frac{\pi}{2}$, Frequency $f=\frac{2}{\pi}$
2. $g(x)=-3\cos(2x)$: Period $T=\pi$, Frequency $f=\frac{1}{\pi}$
3. $h(x)=\cos(\frac{1}{2}x)+2$: Period $T = 4\pi$, Frequency $f=\frac{1}{4\pi}$
4. $k(x)=-2\sin(\frac{3}{4}x)+1$: Period $T=\frac{8\pi}{3}$, Frequency $f=\frac{3}{8\pi}$
5. $j(x)=4\cos(3x)-1$: Period $T=\frac{2\pi}{3}$, Frequency $f=\frac{3}{2\pi}$