Q. 69 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for each graph.

Step-by-Step Solution

Answer

Equation for the graph is:

$y=-\sin \left(\frac{3}{2}x\right)$

Step 1. Inspecting the y-intercept

Referring to the given graph the type of sinusoid can be determined immediately by inspecting its y- intercept if its at a maximum(highest point) and a minimum(lowest point) or neither 

At its y-intercept neither maximum or nor minimum, then the graph is a sine function and the minimum occurs before the maximum hence,

$A<0$

Step 2. Find the amplitude and period

From the graph, $$\begin{gathered} y_{min}=-1 \\ {y}_{max}=1 \end{gathered}$$

$$\begin{gathered} A=-\frac{1}{2}(y_{max}-y_{min}) \\ A=-\frac{1}{2}(1-(-1\left)\right) \\ A=-\frac{1}{2}\left(2\right) \\ A=-1 \\ \left|A\right|=1 \end{gathered}$$

Using the maximum at  $\frac{\mathrm{\pi }}{3}$, the next maximum occurs at $\frac{5\mathrm{\pi }}{3}$

Therefore,

$T=\frac{4\mathrm{\pi }}{3}$

Step 3. Solve for the equation

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ \omega =\frac{2\mathrm{\pi }\times 3}{4\mathrm{\pi }} \\ \omega =\frac{3}{2} \end{gathered}$$

Therefore, the equation of the graph is 

$y=-\sin \left(\frac{3}{2}x\right)$