Chapter 6, Problem 63RE

To determine

Determine the amplitude and period of each function without graphing.

Answer to Problem 63RE

Amplitude of $y=4\cos x$ is $\boxed{amplitude=4}$

Periodicity of $y=4\cos x$ is $\boxed{periodicity=2\pi }$

Explanation of Solution

Given information:

  $y=4\cos x$

Formula used:

For $f\left(x\right)=a.\cos \left(bx-c\right)+d$ the amplitude is $\left|a\right|$

Periodicity of $a\cdot \cos \left(bx+c\right)+d=\frac{\text{Periodicity of }\cos x}{\left|b\right|}$

Now calculating the amplitude of the function by using $f\left(x\right)=a.\cos \left(bx-c\right)+d$ the amplitude is $\left|a\right|$ formula:

  $\begin{array}{l}f\left(x\right)=4\cos x\\ Here,a=4\\ amplitude=\left|a\right|\\ amplitude=\left|4\right|\\ \boxed{amplitude=4}\end{array}$

Now calculating periodicity by using $a\cdot \cos \left(bx+c\right)+d=\frac{\text{Periodicity of }\cos x}{\left|b\right|}$ formula:

  $\frac{\text{Periodicity of }\cos x}{\left|b\right|}$

Periodicity of $\cos x$ is $2\pi$ and $b=1$ , therefore:

  $\begin{array}{l}=\frac{2\pi }{\left|1\right|}\\ \boxed{periodicity=2\pi }\end{array}$