Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

Write the equation of a sine function that has the given characteristics.

Amplitude: $3$

Period: $\frac{\mathrm{\pi }}{2}$

Phase shift:$-\frac{1}{3}$

Step-by-Step Solution

Answer

The obtained equation from the given information is  $y=3\sin \left(\frac{2}{3}x+\frac{2}{9}\right)$

Step 1. Find the value of ϕ and ω

We know that,  $T=\frac{2\mathrm{\pi }}{\omega }$.

So,

Substituting $T=3\mathrm{\pi }$ in the above equation, we get:

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

We have:

$$\begin{gathered} \frac{\varphi }{\omega }=-\frac{1}{3} \\ \frac{\varphi }{{\displaystyle \frac{2}{3}}}=\frac{-1}{3} \\ \varphi =-\frac{2}{9} \end{gathered}$$

Step 2. Find the equation

The general sine function is $y=A\sin \left(\omega x-\varphi \right)$.

Substituting the value $$\begin{gathered} \omega =\frac{2}{3} \\ \varphi =\frac{-1}{3} \end{gathered}$$, we get:

$$\begin{gathered} y=3\sin \left(\frac{2}{3}x-\left(\frac{2}{9}\right)\right) \\ y=3\sin \left(\frac{2}{3}x+\frac{2}{9}\right) \end{gathered}$$