Q 18. - Precalculus Enhanced with Graphing Utilities

Question

Determine the amplitude and period of each function without graphing. 

$y=\frac{4}{3}\sin \left(\frac{2}{3}x\right)$.

Step-by-Step Solution

Answer

The amplitude is $\frac{4}{3}$ and period is $3\mathrm{\pi }$.

Step 1. Given Information.

Determine the amplitude and period of each function without graphing. 

$y=\frac{4}{3}\sin \left(\frac{2}{3}x\right)$.

Step 2. Evaluation.

For $\omega >0$, the amplitude and period of $y=A\sin \left(\omega x\right)$ is given by

Amplitude $=\left|A\right|$ , period $=\frac{2\mathrm{\pi }}{\omega }$.

By comparing the equation , we get

$A=\frac{4}{3} , \omega =\frac{2}{3}$.

Then, $T=\frac{2\mathrm{\pi }}{{\displaystyle \frac{2}{3}}}=3\mathrm{\pi }$.