## Precalculus (6th Edition) Blitzer

The amplitude and period of the function are $2$ and $8\pi$ , respectively.
We have the trigonometric function $y=2\sin \frac{1}{4}x$ The amplitude is the maximum value of $y$. The maximum value of the given trigonometric function is $2$. So, the amplitude of the trigonometric function is $2$. The function after a certain interval starts repeating itself, and this interval is known as the period of the function. And the period $\left( B \right)$ of the trigonometric function is \begin{align} & B=\frac{2\pi }{\left( \frac{1}{4} \right)} \\ & =8\pi \end{align} And the quarter period is $\frac{8\pi }{4}$ or $2\pi$. The cycle begins at $x=0$. Add quarter periods to find out the key points. First key point is ${{x}_{1}}=0$ Second key point is \begin{align} & {{x}_{2}}=0+2\pi \\ & =2\pi \end{align} Third key point is \begin{align} & {{x}_{3}}=2\pi +2\pi \\ & =4\pi \end{align} Fourth key point is \begin{align} & {{x}_{4}}=4\pi +2\pi \\ & =6\pi \end{align} Fifth key point is \begin{align} & {{x}_{5}}=6\pi +2\pi \\ & =8\pi \end{align}