In the standard \((x,y)\) coordinate plane, the graph of the equation \(y=3\sin{(2x+0.5\pi)}\) has what amplitude and period?

$$ \begin{array}{l c c} & \underline{\text{amplitude}} & \underline{\text{period}} \\ \text{A.} & 3 & \pi \\ \text{B.} & 3 & 2\pi \\ \text{C.} & 3 & 4\pi \\ \text{D.} & 6 & \pi \\ \text{E.} & 6 & 2\pi \\ \end{array} $$

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The standard equation for a trigonometric function is:

$$ y=A\sin{B(x-C)}+D $$

The amplitude is \(A\) and the period is related to \(B\) by:

$$ \text{Period} = \frac{2\pi}{B} $$

Rewriting the given equation:

$$ y=3\sin{(2x+0.5\pi)} $$ $$ y = 3\sin{2(x+0.25\pi)} $$

In this form we can see that the amplitude is 3 and the period is:

$$ \text{Period} = \frac{2\pi}{B} $$ $$ \text{Period} = \frac{2\pi}{2} $$ $$ \text{Period} =\pi $$