Chapter 6.4, Problem 33AYU

To determine

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

Answer to Problem 33AYU

Solution:

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

Range: $\left[-4, 4\right]$

Explanation of Solution

Given:

  $y=-4\sin x$

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 $y=3\sin x$ to $y = A\text{sin }\left(\omega x\right)$ , $A=-4$ and $\omega =1$

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

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

Because the amplitude is 4, the graph of $y=-4\sin x$ will lie between $-4$ and $4$ 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 $y=-4\sin x$ , multiply the y-coordinates of the five key points of $y=\sin x$ by $A=-4.$ The five key points are

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

  

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

Range: $\left[-4, 4\right]$