Chapter 4.4, Problem 73E

Solution

To calculate: The amplitude and period of the sinusoid $y=\frac{2}{3}\cos \left(\frac{x-3}{4}\right)+1$ , and the phase shift and vertical translation.

The amplitude of the function is $\frac{2}{3}$ , period is $8\pi$ and the function has a phase shift of 3 and vertical translation of 1 unit1 up.

Given information:

The given function is $y=\frac{2}{3}\cos \left(\frac{x-3}{4}\right)+1$ .

Formula used: The period formula is $\text{Period}=\frac{2\pi }{\left|b\right|}$ .

Calculation:

Consider the given function: $y=\frac{2}{3}\cos \left(\frac{x-3}{4}\right)+1$

The general form of the sinusoid function is: $y=a\cos \left[b\left(x-h\right)\right]+k$

Here $a$ is the amplitude of the graph and $b$ are constant, h is phase shift and k is vertical shift.

Now rewrite the given sinusoid in general form:

  $y=\frac{2}{3}\cos \left[\frac{1}{4}\left(x-3\right)\right]+1$

The values are $a=\frac{2}{3},b=\frac{1}{4},h=3,k=1$ .

The sinusoid function has a phase shift of 3 and vertical translation of 1 unit up.

The amplitude of the function is $\left|a\right|=\frac{2}{3}$ .

Now find the period of the function.

  $\text{Period}=\frac{2\pi }{\left|b\right|}$

Substitute the value $b=\frac{1}{4}$ in this formula.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\left|\frac{1}{4}\right|}\\ =8\pi \end{array}$

Therefore, the amplitude of the function is $\frac{2}{3}$ , period is $8\pi$ and the function has a phase shift of 3 and vertical translation of 1 unit1 up.