Chapter 6.4, Problem 39AYU

To determine

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

Answer to Problem 39AYU

Solution:

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

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

Explanation of Solution

Given:

  $y=2\sin \left(\frac{1}{2}x\right)$

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

Amplitude = $\left|A\right|=2$

Period $T=\frac{2\pi }{\omega }=\frac{2\pi }{\frac{1}{2}}=4\pi$

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

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

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

  $\left(0, 0\right) \left(\pi ,2\right) \left(2\pi ,0\right) \left(3\pi ,-2\right) \left(4\pi , 0\right)$

  

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

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