Chapter 4.5, Problem 37E

To determine

To sketch: the graphs of $f$ and $g$ in the same coordinate plane.

Explanation of Solution

Given information:

Given functions are

  $\begin{array}{l}f\left(x\right)=2\cos x\\ g\left(x\right)=2\cos \left(x+\pi \right)\end{array}$

Concept used:

For the standard equation is $y=a\cos \left(bx-c\right)$ or $y=a\cos bx$

Amplitude is $\left|a\right|$ and period is $\frac{2\pi }{b}$ .

Vertical shift is $d$ unit and horizontal shift is $\frac{c}{b}$

Calculation:

Comparing $f\left(x\right)=2\cos x$ with the equation $y=a\cos bx$ ,

  $b=1,a=2$

So,

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

Comparing $g\left(x\right)=2\cos \left(x+\pi \right)$ with the equation $y=a\cos \left(bx-c\right)$ ,

  $b=1,a=2$

So,

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

The two curves has same period and same amplitude, but they will be shifted horizontal by $\pi$ units to the left. The graphs of $f\left(x\right)$ and $g\left(x\right)$ is given below: