Question

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

Amplitude: $2$

Period: $\mathrm{\pi }$

Phase shift:$\frac{1}{2}$

Short answer

The obtained equation from the given information is$y=2\sin \left(2x-1\right)$

Step-by-step solution

Step 1. Find the value of ω and ϕ

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

So,

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

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

We have:,

$$\begin{gathered} \frac{\varphi }{\omega }=\frac{1}{2} \\ \frac{\varphi }{2}=\frac{1}{2}\left[\omega =2\right] \\ \varphi =1 \end{gathered}$$

Step 2. Writing the equation

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

Substituting $$\begin{gathered} A=2 \\ \omega =2 \\ \varphi =1 \end{gathered}$$in the above equation, we get:

$y=2\sin \left(2x-1\right)$