Chapter 6.4, Problem 31AYU

To determine

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

Answer to Problem 31AYU

Solution:

Domain: (-∞, ∞)Range: [-4, 4].

Explanation of Solution

Given:

y = 4 cos x

Formula used:

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

Calculation:

If ω>0, the amplitude and period of y = A sin (ωx) and y = A cos (ωx) are given by

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

Comparing y = 4 cos x to y = A cos (ωx) , A = 4 and ω=1.

Amplitude = ω = 4 .

Period $T=\frac{2\pi }{\omega }=\frac{2\pi }{1}=2\pi$.

Because the amplitude is 4, the graph of y = 4 cos x will lie between -4 and 4 on the y-axis. Because the period is 2π, one cycle will begin at x = 0 and end at x = 2π.

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

To obtain the y-coordinates of the five key points of y = 4 cos x, multiply the y-coordinates of the five key points of y = cos x by A = 4. The five key points are $(0,4)\left(\frac{\pi }{2},0\right)(\pi ,-4)\left(\frac{3\pi }{2},0\right)(2\pi ,4)$.

  

Domain: (-∞, ∞)Range: [-4, 4].