Chapter 9.6, Problem 16E

To determine

To find: A function for given sinusoidal graph.

Answer to Problem 16E

  $y=-\sin \left(\frac{4\pi }{3}t\right)-2$

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 $-1$ and minimum value is $-3$ .

Step 2: Identify the vertical shift $k$ .

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

  $k=\frac{-1+\left(-3\right)}{2}=-2$

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

Since at $t=0$ , graph is at midpoint and going down. So, use sine function reflected across $x-$ axis with no horizontal shift.

Step 4: Find the amplitude and period.

Since the graph repeats itself every $\frac{3}{2}$ unit. Period is $\frac{3}{2}$ .

  $\text{i}\text{.e}\text{. }\frac{2\pi }{b}=\frac{3}{2}\Rightarrow b=\frac{4\pi }{3}$ .

  $\begin{array}{l}\left|a\right|=\frac{\text{Maximum value}-\text{Minimum value}}{2}=\frac{-1-\left(-3\right)}{2}=1\\ \text{Since graph is a reflection, }a<0\\ \Rightarrow a=-1\end{array}$

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

  $\begin{array}{l}y=-1\sin \left(\frac{4\pi }{3}t\right)-2\\ \Rightarrow y=-\sin \left(\frac{4\pi }{3}t\right)-2\end{array}$

Hence, desired function of the given graph is $y=-\sin \left(\frac{4\pi }{3}t\right)-2$