Chapter 14, Problem 8CLR

To determine

To find: The amplitude, the maximum and minimum values, and the period of the function $y=5\cos \frac{\pi }{3}x-2$ .

Answer to Problem 8CLR

The amplitude is $5$ , period is $6$ , maximum value is $3$ and minimum value is $-7$ .

Explanation of Solution

Given:

The function is $y=5\cos \frac{\pi }{3}x-2$ .

Calculation:

For positive real numbers $a$ and $b$ , the function $y=a\cos bx+c$ have amplitude $a$ and period $\frac{2\pi }{b}$ and has maximum value $M=c+a$ and minimum value $m=c-a$ .

Compare the function $y=5\cos \frac{\pi }{3}x-2$ with the general form $y=a\cos bx+c$ .

  $\begin{array}{c}a=5\\ b=\frac{\pi }{3}\\ c=-2\end{array}$

Find the period as follows.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{b}\\ =\frac{2\pi }{\frac{\pi }{3}}\\ =6\end{array}$

Find the maximum value as follows.

  $\begin{array}{c}M=c+a\\ =-2+5\\ =3\end{array}$

Find the minimum value as follows.

  $\begin{array}{c}m=c-a\\ =-2-5\\ =-7\end{array}$

Therefore, the amplitude is $5$ , period is $6$ , maximum value is $3$ and minimum value is $-7$ .