Chapter 9.6, Problem 15E

To determine

To find: A function for given sinusoidal graph.

Answer to Problem 15E

  $y=-2\cos \left(\frac{\pi }{2}t\right)$

Explanation of Solution

Given information:

A sinusoidal graph is given as below.

  

Calculation:

A sinusoidal graph is given as below.

  

Step 1: Find the maximum and minimum value.

From above graph, maximum value is $2$ and minimum value is $-2$ .

Step 2: Identify the vertical shift $k$ .

The value of $k$ is the mean of maximum and minimum value, i.e.

  $k=\frac{2+\left(-2\right)}{2}=0$

Step 3: Decide whether graph should be modeled as sine or cosine function.

At $t=0$ , graph is at its lowest. So, use cosine function reflection in $x-$ axis with no horizontal shift.

Step 4: Find the amplitude and period.

Since the graph repeats itself every $4$ unit. Period is $4$ .

  $\text{i}\text{.e}\text{. }\frac{2\pi }{b}=4\Rightarrow b=\frac{\pi }{2}$ .

  $\begin{array}{l}\left|a\right|=\frac{\text{Maximum value}-\text{Minimum value}}{2}=\frac{2-\left(-2\right)}{2}=2\\ \text{Since graph is a reflection, }a<0\\ \Rightarrow a=-2\end{array}$

Step $5$ : The function graph would be given as −

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

Hence, desired function of the given graph is $y=-2\cos \left(\frac{\pi }{2}t\right)$