Chapter 4, Problem 1RE

Concept Check Which one of the following statements is true about the graph of y = 4 sin 2x?

A. It has amplitude 2 and period $\frac{\pi }{2}$.

B. It has amplitude 4 and period $\pi$.

C. Its range is [0,4].

D. Its range is [-4. 0].

To determine

The correct statement about the graph of the given expression from the given options.

Answer to Problem 1RE

Solution: B

Given: The expression $y=4\sin 2x$ and the options:

$\begin{array}{l}A.\text{ It has amplitude 2 and period }\frac{\pi }{2}\\ B.\text{ It has amplitude 4 and period }\pi \\ C.\text{ Its range is }\left[0,4\right]\\ D.\text{ Its range is }\left[-4,0\right]\end{array}$

Explanation of Solution

In the general form of the function $y=a\sin bx$, $\left|a\right|$ represents the amplitude of the function and the period of the sine function is $\frac{2\pi }{b}$ as

$\begin{array}{c}0\le bx\le 2\pi \\ 0\le x\le \frac{2\pi }{b}\end{array}$

Now, in the mentioned function $y=4\sin 2x$, the amplitude is 4 and the period is $\frac{2\pi }{2}=\pi$. The range of $a\sin bx$ is the interval $\left[-a,a\right]$. Thus, the range of provided function is $\left[-4,4\right]$.

Hence, the correct option is B.

The given function has an amplitude of 4, period of $\pi ,$ and range of $\left[-4,4\right]$.