Q. 75 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for the graph.

Step-by-Step Solution

Answer

The equation of the given graph is $y=-4\sin \left(3x\right)$.

Step 1. Determine the sinusoidal function.

Consider 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.

As the minimum is at its y-intercept, then the graph is a cosine function and $A<0$. Then,

$$\begin{gathered} y_{max}=4 \\ {y}_{min}=-4 \end{gathered}$$

Step 2. Find the amplitude and period.

Consider the given question,

$$\begin{gathered} A=-\frac{1}{2}\left(y_{max}-y_{min}\right) \\ A=-\frac{1}{2}\left[4-\left(-4\right)\right] \\ A=-\frac{1}{2}\left(4+4\right) \\ A=-\frac{1}{2}\times 8 \\ A=-4 \end{gathered}$$

The period is $\frac{2\mathrm{\pi }}{3}$. So,

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

Step 3. Form the equation.

Vertical shift,

$$\begin{gathered} b=\frac{1}{2}\left(y_{max}+y_{min}\right) \\ b=\frac{1}{2}\left(4-4\right) \\ b=\frac{1}{2}\left(0\right) \\ b=0 \end{gathered}$$

Consider the form of the function,

$y=A\sin \left(\omega x\right)+b$

Substitute the values,

$$\begin{gathered} y=-4\sin \left(3x\right)+0 \\ y=-4\sin \left(3x\right) \end{gathered}$$