Chapter 6.4, Problem 54AYU

In Problems 33-56, graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function.

$y=-\frac{1}{2}\sin \left(\frac{\pi }{8}x\right)+\frac{3}{2}$

To determine

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

Answer to Problem 54AYU

Domain: $\left(-\infty , \infty \right)$ Range: $\left[0.5, 1\right]$.

Explanation of Solution

Given:

$y=-\frac{1}{2}\sin \left(\frac{\pi }{8}x\right)+\frac{3}{2}$

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: $y=\frac{1}{2}\sin \left(\frac{\pi }{8}x\right)$ to $y =A\text{sin}\left(\omega x\right)$, $A=-\frac{1}{2}$ and $\omega =\frac{\pi }{8}$.

Amplitude $=\left|A\right|=\frac{1}{2}$.

Period $T=\frac{2\pi }{\omega }=\frac{2\pi }{\frac{\pi }{3}}=16$.

Because the amplitude is $\frac{1}{2}$, the graph of $y=-\frac{1}{2}\sin \left(\frac{\pi }{8}x\right)$ will lie between $-\frac{1}{2}$ and $\frac{1}{2}$ on the $y\text{-axis}$. Because the period is $16$, one cycle will begin at $x = 0$ and end at $x =16$.

Divide the interval $\left[0,16\right]$ into four subintervals, each of length $\frac{16}{4}=4$. The $x-\text{coordinates}$ of the five key points are $0,4,8,12,16$.

To obtain the $y-\text{coordinates}$ of the five key points of: $y=-\frac{1}{2}\sin \left(\frac{\pi }{8}x\right)$, multiply the $y-\text{coordinates}$ of the five key points of $y=\sin x$ by $A=-\frac{1}{2}$. The five key points are

$\left(0, 0\right) \left(4,-\frac{1}{2}\right) \left(8, 0\right) \left(12, \frac{1}{2}\right) \left(16, 0\right)$

A vertical shift up $\frac{3}{2}$ units gives the graph of $y=-\frac{1}{2}\cos \left(\frac{\pi }{8}x\right)+ \frac{3}{2}$.

Domain: $\left(-\infty , \infty \right)$ Range: $\left[0.5,1\right]$.