examples 3 and 4\nfind the amplitude and period of each function. then graph the function.\n11. $y =…

Explanation:

Step1: Recall amplitude and period formulas

For a sine or cosine function of the form $y = A\sin(B\theta)$ or $y=A\cos(B\theta)$, the amplitude is $|A|$ and the period is $\frac{2\pi}{|B|}$.

Step2: Solve for $y = 3\sin\theta$

Amplitude: $|A|=|3| = 3$. Period: Since $B = 1$, $\frac{2\pi}{|B|}=\frac{2\pi}{1}=2\pi$.

Step3: Solve for $y=\cos3\theta$

Amplitude: $|A| = |1|=1$. Period: Since $B = 3$, $\frac{2\pi}{|B|}=\frac{2\pi}{3}$.

Step4: Solve for $y=\sin4\theta$

Amplitude: $|A|=|1| = 1$. Period: Since $B = 4$, $\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2}$.

Step5: Solve for $y=\frac{3}{2}\sin\theta$

Amplitude: $|A|=\left|\frac{3}{2}\right|=\frac{3}{2}$. Period: Since $B = 1$, $\frac{2\pi}{|B|}=2\pi$.

Step6: Solve for $y = 4\cos2\theta$

Amplitude: $|A|=|4| = 4$. Period: Since $B = 2$, $\frac{2\pi}{|B|}=\frac{2\pi}{2}=\pi$.

Step7: Solve for $y = 5\sin\frac{2}{3}\theta$

Amplitude: $|A|=|5| = 5$. Period: Since $B=\frac{2}{3}$, $\frac{2\pi}{|B|}=\frac{2\pi}{\frac{2}{3}}=3\pi$.

Answer:

11. Amplitude: 3, Period: $2\pi$
12. Amplitude: 1, Period: $\frac{2\pi}{3}$
13. Amplitude: 1, Period: $\frac{\pi}{2}$
14. Amplitude: $\frac{3}{2}$, Period: $2\pi$
15. Amplitude: 4, Period: $\pi$
16. Amplitude: 5, Period: $3\pi$