Chapter 10, Problem 10T

Solution

To find: The function for the sinusoid.

  $y=3\cos \pi x+2$

Given information:

Given, sinusoid is:

  

Calculation:

Let the function be of the form $y=a\cos b\left(x-h\right)+k$

For a function of the form $y=a\cos b\left(x-h\right)+k$ , a is the amplitude, $\frac{2\pi }{b}$ is the period, h is the horizontal shift and k is the vertical shift.

From the graph,

The amplitude is given by $\frac{5-\left(-1\right)}{2}=\frac{5+1}{2}=\frac{6}{2}=3$

The period of the graph is $2$

The horizontal shift is 0.

The vertical shift is midpoint of the maximum and minimum and is given by $\frac{5-1}{2}=\frac{4}{2}=2$ .

Hence,

  $\begin{array}{l}\text{Amplitude: }a=3\\ \text{Period:}\frac{2\pi }{b}=2\Rightarrow b=\pi \\ \text{Horizontal Shift}=0\\ \text{Vertical Shift}=2\end{array}$

Here, $a=3,b=\pi ,h=0,k=2$

Therefore, the equation is:

  $\begin{array}{l}y=3\cos \pi \left(x-0\right)+2\\ y=3\cos \pi x+2\end{array}$

Conclusion:

  $y=3\cos \pi x+2$