Chapter 6.4, Problem 35AYU

To determine

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

Answer to Problem 35AYU

Solution:

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

Range: $\left[-1, 1\right]$

Explanation of Solution

Given:

  $y=\cos \left(4x\right)$

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 $y=\cos \left(4x\right)$ to $y = A\text{cos }\left(\omega x\right)$ , $A=1$ and $\omega =4$

Amplitude = $\left|A\right|=1$

Period $T=\frac{2\pi }{\omega }=\frac{2\pi }{4}=\frac{\pi }{2}$

Because the amplitude is 1, the graph of $y=\cos \left(4x\right)$ will lie between $-1$ and $1$ on the $y$ -axis. Because the period is $\frac{\pi }{2}$ , one cycle will begin at $x = 0$ and end at $x =\frac{\pi }{2}$ .

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

To obtain the y-coordinates of the five key points of $y=\cos 4x$ , multiply the y-coordinates of the five key points of $y=\cos x$ by $A=1.$ The five key points are

  $\left(0, 1\right) \left(\frac{\pi }{8},0\right) \left(\frac{\pi }{4},-1\right) \left(\frac{3\pi }{8},0\right) \left(\frac{\pi }{2}, 1\right)$

  

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

Range: $\left[-1, 1\right]$