Chapter 8, Problem 1RE

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

1. $f\left(x\right)=-3\cos x+3$

To determine

To graph two full periods of each function and state the amplitude, period, and midline equation and asymptotes.

Answer to Problem 1RE

  

The amplitude, period, and midlineare $3$, $2\pi$ and $y=3$ respectively.

Explanation of Solution

Given information:

Given function is $f\left(x\right)=-3\cos x+3$.

Calculation:

For a function of the form $f\left(x\right)=A\cos \left(Bx-C\right)+D$ :

The amplitude is given by $\left|A\right|$

The period is given by $P=\frac{2\pi }{\left|B\right|}$

The midline is given by $y=D$

For a Cosine function, the asymptotes does not exist.

Here, $A=-3,B=1,C=0,D=3$

Plugging the values in the formulae,

  $\text{Amplitude}=\left|A\right|=\left|-3\right|=3$

Period of the graph is

  $P=\frac{2\pi }{\left|B\right|}=\frac{2\pi }{\left|1\right|}=\frac{2\pi }{1}=2\pi$

Midline of the graph is

  $\begin{array}{l}y=D\\ y=3\end{array}$

Graph:

Graphing two periods of the function using graphing calculator:

  

Therefore, the amplitude, period, and midlineare $3$, $2\pi$ and $y=3$ respectively and asymptotes do not exist.