Question

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

Amplitude:$3$

Period:$\pi$

Phase shift:$2$

Short answer

The obtained equation from the given information is $y=3\sin 2\left(x-4\right)$

Step-by-step solution

Step 1.  Find the value of ω and ϕ 

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

So,

Substituting $T=\frac{\mathrm{\pi }}{2}$in the above equation, we get:

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

We have:

$$\begin{gathered} \frac{\varphi }{\omega }=2 \\ \frac{\varphi }{4}=2\left[\omega =4\right] \\ \varphi =8 \end{gathered}$$

Step 2. Writing the equation 

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

Substituting $$\begin{gathered} A=3 \\ \omega =4 \\ \varphi =8 \end{gathered}$$in the above equation, we get:

$$\begin{gathered} y=3\sin \left(4x-8\right) \\ y=3\sin 4\left(x-2\right) \end{gathered}$$