Q 20. - Precalculus Enhanced with Graphing Utilities

Question

Determine the amplitude and period of each function without graphing. 

$y=\frac{9}{5}\cos \left(-\frac{3\mathrm{\pi }}{2}x\right)$.

Step-by-Step Solution

Answer

The amplitude is $\frac{9}{5}$ and period is $\frac{4}{3}$.

Step 1. Given Information.

Determine the amplitude and period of each function without graphing. 

$y=\frac{9}{5}\cos \left(-\frac{3\mathrm{\pi }}{2}x\right)$.

Step 2. Evaluation.

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

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

On comparing, we get

Amplitude $=\left|\frac{9}{5}\right|=\frac{9}{5}$ , $\omega =-\frac{3\mathrm{\pi }}{2}$

Then, Period $=T=\frac{2\mathrm{\pi }}{-{\displaystyle \frac{3\mathrm{\pi }}{2}}}=-\frac{4}{3}$.