Q. 73 - Precalculus Enhanced with Graphing Utilities

Question

Find an equation for the graph.

Step-by-Step Solution

Answer

The equation of the given graph is $y=3\sin \left(\frac{\mathrm{\pi }}{2}x\right)$.

Step 1. Determine the sinusoidal function.

Consider the given graph,

The type of sinusoidal graph can be determined immediately by inspecting its y-intercept if its at a maximum(highest point) and a minimum(lowest point) or neither.

As there is neither a maximum or minimum at its y-intercept, then the graph is a sine function and the minimum occurs before the maximum and $A>0$. Then,

$$\begin{gathered} y_{max}=3 \\ {y}_{min}=-3 \end{gathered}$$

Step 2. Find the amplitude and period.

Consider the given question,

$$\begin{gathered} A=\frac{1}{2}\left(y_{max}-y_{min}\right) \\ A=\frac{1}{2}\left[3-\left(-3\right)\right] \\ A=\frac{1}{2}\left(6\right) \\ A=3 \end{gathered}$$

The period is $4$. So,

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

Step 3. Form the equation.

Vertical shift,

$$\begin{gathered} b=\frac{1}{2}\left(y_{max}+y_{min}\right) \\ b=\frac{1}{2}\left(3-3\right) \\ b=0 \end{gathered}$$

Consider the form of the function,

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

Substitute the values,

$$\begin{gathered} y=3\sin \left(\frac{\mathrm{\pi }}{2}x\right)+0 \\ y=3\sin \left(\frac{\mathrm{\pi }}{2}x\right) \end{gathered}$$