## Precalculus (6th Edition) Blitzer

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