Chapter 6.4, Problem 41AYU

To determine

To find: The graph of each function using method of key points. Also find domain and range.

Answer to Problem 41AYU

Solution:

Domain: $\left(-\infty , \infty \right)$

Range: $\left[-\frac{1}{2},\frac{1}{2}\right]$

Explanation of Solution

Given:

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

Calculation:

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

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

Comparing : $y=-\frac{1}{2}\cos \left(2x\right)$ to $y = A\text{cos }\left(\omega x\right)$ , $A=-\frac{1}{2}$ and $\omega =2$

Amplitude = $\left|A\right|=\frac{1}{2}$

Period $T=\frac{2\pi }{\omega }=\frac{2\pi }{2}=\pi$

Because the amplitude is 1, the graph of : $y=-\frac{1}{2}\cos \left(2x\right)$ will lie between $-\frac{1}{2}$ and $\frac{1}{2}$ on the $y$ -axis. Because the period is $\pi$ , one cycle will begin at $x = 0$ and end at $x =\pi$ .

Divide the interval $\left[0, 8\pi \right]$ into four subintervals, each of length $\frac{\pi }{4}=\frac{\pi }{4}$ . The x-coordinates of the five key points are: $0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi$

To obtain the y-coordinates of the five key points of : $y=-\frac{1}{2}\cos \left(2x\right)$ , multiply the y-coordinates of the five key points of $y=\cos x$ by $A=-\frac{1}{2}.$ The five key points are

  $\left(0, -\frac{1}{2}\right) \left(\frac{\pi }{4},0\right) \left(\frac{\pi }{2},\frac{1}{2}\right) \left(\frac{3\pi }{4},0\right) \left(\pi ,-\frac{1}{2}\right)$

  

Domain: $\left(-\infty , \infty \right)$

Range: $\left[-\frac{1}{2},\frac{1}{2}\right]$