Chapter 9.4, Problem 14E

To determine

To identify: The amplitude and period of the function then graph the function and describe the function of $g$ as a transformation of the graph of its parent function.

Answer to Problem 14E

The amplitude is 2 and period is $2\pi$ and graph is shown in figure (1).

Explanation of Solution

Given:

The given function is $g\left(x\right)=2\sin x$ .

Calculation:

Compare the function $g\left(x\right)=2\sin x$ with $y=a\sin b\left(x-h\right)+k$ to find amplitude and period of the function.

  $\begin{array}{c}a=2\\ \text{Period}=\frac{2\pi }{b}\\ =\frac{2\pi }{1}\\ =1\end{array}$

The horizontal shift is $h=0$ and vertical shifts is $k=0$ .

Draw the midline of the graph, $y=0$ .

Find the intercepts at $x=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$ to draw the graph.

  $\begin{array}{c}\left(0,2\cdot 0\right)=\left(0,0\right)\\ \left(\frac{\pi }{2},2\cdot 1\right)=\left(\frac{\pi }{2},2\right)\\ \left(\pi ,2\cdot 0\right)=\left(\pi ,0\right)\\ \left(\frac{3\pi }{2},2\cdot \left(-1\right)\right)=\left(\frac{3\pi }{2},-2\right)\end{array}$

And,

  $\left(2\pi ,2\cdot 0\right)=\left(2\pi ,0\right)$

Maximum point is $\left(\frac{\pi }{2},2\right)$ and minimum point is $\left(\frac{3\pi }{2},-2\right)$ .

Draw the graph of the function $g\left(x\right)=2\sin x$ as follows.

  

Figure (1)

The graph of $g$ is vertical stretch by factor of 2 of the graph of $f$ .

Therefore, the amplitude is 2 and period is $2\pi$ and graph is shown in figure (1).