Q. 18 - Precalculus Enhanced with Graphing Utilities

Question

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

Amplitude: $2$

Period: $\mathrm{\pi }$

Phase shift:$-2$

Step-by-Step Solution

Answer

The obtained equation from the given information is $y=2\sin \left(2x+4\right)$

Step 1. Find the value of ω and ϕ

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

So,

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

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

We have:

$$\begin{gathered} \frac{\varphi }{\omega }=-2 \\ \frac{\varphi }{2}=-2 \\ \varphi =-4 \end{gathered}$$

Step 2. Find the equation

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

Substituting the value $$\begin{gathered} \omega =2 \\ \varphi =-4 \\ A=2 \end{gathered}$$ in the general sine function, we get:

$$\begin{gathered} y=2\sin \left(2x-(-4)\right) \\ y=2\sin \left(2x+4\right) \end{gathered}$$