Chapter 9.4, Problem 15Q

To determine

To find: The amplitude and period of the given function and draw the graph.

Answer to Problem 15Q

The amplitude and period and graph of the function are given below.

Explanation of Solution

Given information:

The given function $g$ is:

  $g\left(x\right)=\cos 5\pi x+3$ .

And the function $f$ is:

  $f\left(x\right)=\cos x$

The function is of the form $g\left(x\right)=a\cos \left(b-h\right)x+k$ , here $a=1\text{and}b=5\pi$ . So the amplitude is $a=1$ and the period is:

  $\frac{2\pi }{5\pi }=\frac{2}{5}$

The table for the graph is given as:

$x$	$0$	$\frac{1}{5}$	$\frac{1}{10}$	$\frac{-1}{5}$	$\frac{-1}{10}$
$g\left(x\right)$	$4$	$2$	$3$	$2$	$3$

The graph of the function is given as:

  

The description about the graph of $g$ as a transformation of the graph $f$ is given as:

The graph of $g$ is horizontally shrinks by a factor of $5\pi$ and a translation by $3$ units upward of the graph of $f$ .

Therefore, the amplitude and period and graph of the function are given above.