Chapter 6.4, Problem 50AYU

To determine

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

Answer to Problem 50AYU

Solution:

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

Range: $\left[-2, 6\right]$

Explanation of Solution

Given:

  $y=2-4\cos \left(3x\right)$

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: $y=-4\cos \left(3x\right)$ to $y = A\text{cos }\left(\omega x\right)$ , $A=-4$ and $\omega =3$

Amplitude = $\left|A\right|=4$

Period $T=\frac{2\pi }{\omega }=\frac{2\pi }{3}=\frac{2\pi }{3}$

Because the amplitude is 4, the graph of $y=-4\cos \left(3x\right)$ will lie between $-4$ and $4$ on the $y$ -axis. Because the period is $\frac{2\pi }{3}$ , one cycle will begin at $x = 0$ and end at $x =\frac{2\pi }{3}$ .

Divide the interval $\left[0, \frac{2\pi }{3}\right]$ into four subintervals, each of length $\frac{\frac{2\pi }{3}}{4}=\frac{\pi }{6}$ . The x-coordinates of the five key points are: $0, \frac{\pi }{6},\frac{\pi }{3},\frac{3\pi }{6},\frac{2\pi }{3}$

To obtain the y-coordinates of the five key points of $y=-4\cos \left(3x\right)$ , multiply the y-coordinates of the five key points of $y=\cos x$ by $A=-4.$ The five key points are

  $\left(0, -4\right) \left(\frac{\pi }{6},0\right) \left(\frac{\pi }{3},4\right) \left(\frac{3\pi }{6},0\right) \left(\frac{2\pi }{3}, -4\right)$

A vertical shift up 2 units gives the graph of $y=2-4\cos \left(3x\right)$

  

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

Range: $\left[-2, 6\right]$