Q. 27 - Precalculus Enhanced with Graphing Utilities

Question

Write an equation for a sinusoidal graph with the following properties: 

A $=-3$  period $=\frac{2\mathrm{\pi }}{3}$ phase shift  $=-\frac{\mathrm{\pi }}{4}$

Step-by-Step Solution

Answer

The equation is $y=-3\sin \left(3x+\frac{3\mathrm{\pi }}{4}\right)$.

Step 1. Given Information

We are given A $=-3$ period $=\frac{2\mathrm{\pi }}{3}$ phase shift $=-\frac{\mathrm{\pi }}{4}$ and we have to find the equation of the sinusoidal graph.

Step 2. Solve for ω

If $\omega >0$ , the equation of the sinusoidal graph with amplitude $\left|A\right|$, period $T=\frac{2\mathrm{\pi }}{\omega }$ and phase shift $=\frac{\varnothing }{\omega }$ is $y=A\sin \left(\omega x-\varnothing \right)$.

We have, $T=\frac{2\mathrm{\pi }}{3}$

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

Step 3. Solve for ∅

Substituting the value of $\omega =3$ in the equation of phase shift.

We get, 

$$\begin{gathered} \frac{\varnothing }{\omega }=-\frac{\mathrm{\pi }}{4} \\ \Rightarrow \frac{\varnothing }{3}=-\frac{\mathrm{\pi }}{4} \\ \Rightarrow \varnothing =-\frac{\mathrm{\pi }}{4}\times 3 \\ \Rightarrow \varnothing =-\frac{3\mathrm{\pi }}{4} \end{gathered}$$

Step 4. Finding the equation

Substituting all the values in the equation of the graph we get,

$$\begin{gathered} y=A\sin (\omega x-\varnothing ) \\ \Rightarrow y=-3\sin \left(3x-\left(-\frac{3\mathrm{\pi }}{4}\right)\right) \\ \Rightarrow y=-3\sin \left(3x+\frac{3\mathrm{\pi }}{4}\right) \end{gathered}$$