Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 4 - Section 4.5 - Graphs of Sine and Cosine Functions - Exercise Set - Page 595: 10


The amplitude and period of the function are $2$ and $8\pi $ , respectively.

Work Step by Step

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 $\begin{align} & {{x}_{5}}=6\pi +2\pi \\ & =8\pi \end{align}$
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