Chapter 4.5, Problem 28E

To determine

To describe: the relationship between the graphs of $f$ and $g$ .

Explanation of Solution

Given information:

Given graph is

  

Calculation:

The general equation of cosine is,

  $y=a\cos \left(bx-c\right)$

Here $a$ is amplitude, $b$ is angular frequency and $c$ horizontal translation.

Consider the given graph below,

  

Here red color graph is of $f$ and blue color graph is of $g$ .

From the above graph, this can be seen that amplitude is same for both function $f$ and $g$ .

The frequency or time period can be calculated by counting the number of times, the graph is repeated in the interval $\left[-\pi ,\pi \right]$ or $2\pi$ .

The graph $f$ is repeating only 1 time in interval $\left[-\pi ,\pi \right]$ , Thus the value of $b$ is,

  $\begin{array}{c}b=2\pi \left(1\right)\\ =2\pi \end{array}$

As there is no horizontal translation, thus , $c=0$

The equation of function $f$ is,

  $f=\cos \left(2\pi x\right)$

As the function is repeating 3 times, thus the value of $b$ for function $g$ is,

  $\begin{array}{c}b=2\pi \left(3\right)\\ =6\pi \end{array}$

As there is no horizontal translation, thus, $c=0$ .

The equation of function $g$ is,

  $g=\cos \left(6\pi x\right)$

The relationship between $f$ and $g$ is that the graph $f$ has a horizontal stretching with respect to graph $g$ or scaling of 3 times is done in $x$ domain of function $f$ .