Chapter 6.4, Problem 44AYU

To determine

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

Answer to Problem 44AYU

Solution:

Domain:

  $\left(-\infty , \infty \right)$

Range: $\left[-1, 5\right]$

Explanation of Solution

Given:

  $y=3\cos x+2$

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: $y=3\cos x$ to $y = A\text{cos }\left(\omega x\right)$ , $A=3$ and $\omega =1$

Amplitude = $\left|A\right|=3$

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

Because the amplitude is 3, the graph of $y=3\cos x$ will lie between $-3$ and $3$ on the $y$ -axis. Because the period is $2\pi$ , one cycle will begin at $x = 0$ and end at $x = 2\pi$ .

Divide the interval $\left[0, 2\pi \right]$ 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 $y=3\cos x$ , multiply the y-coordinates of the five key points of $y=\cos x$ by $A=3.$ The five key points are

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

A vertical shift up 2 units gives the graph of $y=3\cos x+2$

  

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

Range: $\left[-1, 5\right]$