Chapter 6.6, Problem 11AYU

In Problems 3-14, find the amplitude, period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods.

$y\text{ = }3\text{cos}\left(\pi x-2\right)+5$

To determine

To find: The amplitude, period, phase shift of the function, graph the function.

Answer to Problem 11AYU

Solution:

The amplitude $=2$, period $=2$, phase shift $=\frac{2}{\pi }$.

Explanation of Solution

Given:

$y=3\cos \left(\pi x-2\right)+5$

Calculation:

Compare $y=3\cos \left(\pi x-2\right)+5$ to $y=A\cos \left(\omega x-\varphi \right)+B$, note that $A=3, \omega =\pi$, $\varphi =2$ and $B=5$. The graph is a cosine curve with amplitude $\left|A\right|=3$, period $T=\frac{2\pi }{\omega }=\frac{2\pi }{\pi }=2$ and phase shift $=\frac{\varphi }{\omega }=\frac{2}{\pi }=\frac{2}{\pi }$.

The graph of $y=3\cos \left(\pi x-2\right)+5$ will lie between $-3$ and 3 on the $y\text{-axis}$.

One cycle will begin at $x=\frac{\varphi }{\omega }=\frac{2}{\pi }$ and end at $x=\frac{\varphi }{\omega }+\frac{2\pi }{\omega }=\frac{2}{\pi }+2=\frac{2+2\pi }{\pi }$.

To find five key points, divide the interval $\left[\frac{2}{\pi },\frac{2+2\pi }{\pi }\right]$ in to four sub intervals, each of length $\frac{\frac{2+2\pi }{\pi }-\frac{2}{\pi }}{4}=\frac{2}{4}=\frac{1}{2}$.

Use the values of $x$ to determine the five key points on the graph:

$x$	$y=3\cos \left(\pi x-2\right)$
$\frac{2}{\pi }$	$3$
$\frac{4+\pi }{2\pi }$	$0$
$\frac{4+2\pi }{2\pi }$	$-3$
$\frac{4+3\pi }{2\pi }$	$0$
$\frac{2+2\pi }{\pi }$	$3$

$x$	$y=3\cos \left(\pi x-2\right)+5$
$\frac{2}{\pi }$	$8$
$\frac{4+\pi }{2\pi }$	$5$
$\frac{4+2\pi }{2\pi }$	$2$
$\frac{4+3\pi }{2\pi }$	$5$
$\frac{2+2\pi }{\pi }$	$8$

Plot these five points and fill in the graph of the sine function.

$y=3\cos \left(\pi x-2\right)$	$y=3\cos \left(\pi x-2\right)+5$