Q 17. - Precalculus Enhanced with Graphing Utilities

Question

Determine the amplitude and period of each function without graphing. 

$y=-\frac{1}{2}\cos \left(\frac{3}{2}x\right)$.

Step-by-Step Solution

Answer

Amplitude is $\frac{1}{2}$ and period is $\frac{4\mathrm{\pi }}{3}$.

Step 1. Given Information.

Determine the amplitude and period of each function without graphing. 

$y=-\frac{1}{2}\cos \left(\frac{3}{2}x\right)$.

Step 2. Evaluation.

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

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

After comparing the equation, we get

$A=\left|-\frac{1}{2}\right|=\frac{1}{2}$ , $\omega =\frac{3}{2}$

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