Chapter 9, Problem 12CR

To determine

The amplitude and period of function $y=-4\cos \left(\pi x\right)$ without using the graph.

Answer to Problem 12CR

Solution:

The amplitude of the function $y=-4\cos \left(\pi x\right)$ is $4$ and period is $2$ .

Explanation of Solution

Given information:

The function, $y=-4\cos \left(\pi x\right)$

If $\omega >0$ then the amplitude, period and phase shift of $y=A\sin \left(\omega x-\varphi \right)$ and $y=A\cos \left(\omega x-\varphi \right)$ are given by Amplitude is $\left|A\right|$ , Period $=T=\frac{2\pi }{\omega }$ and phase shift $=\frac{\varphi }{\omega }$ .

Comparing the given function $y=-4\cos \left(\pi x\right)$ to $y=A\cos \left(\omega x-\varphi \right)$ ,

  $\Rightarrow A=-4,\omega =\pi$ and $\varphi =0$

Thus the amplitude is $\left|A\right|=\left|-4\right|=4$ , and period is $T=\frac{2\pi }{\pi }=2$ .