Chapter 4.5, Problem 26E

To determine

To sketch: the graphs of $f$ and $g$ in the same coordinate plane.

Explanation of Solution

Given information:

Given functions are

  $\begin{array}{l}f\left(x\right)=\sin x\\ g\left(x\right)=4\sin x\end{array}$

Concept used:

For the standard equation is $y=a\sin bx$ .

Amplitude is $\left|a\right|$ and period is $\frac{2\pi }{b}$ .

Calculation:

Consider the functions $f\left(x\right)=\sin x$ and $g\left(x\right)=4\sin x$ .

Consider the functions $f\left(x\right)=\sin x$ and $g\left(x\right)=4\sin x$ .

The amplitude of the function $f\left(x\right)=\sin x$ is 1 and amplitude of the function $g\left(x\right)=4\sin x$ is 4. The period of the function $f\left(x\right)=\sin x$ is $2\pi$ and period of function $g\left(x\right)=4\sin x$ is $2\pi$ .

Divide the period interval $\left[0,2\pi \right]$ into four equal parts using the values $\frac{\pi }{2},\pi ,\frac{3\pi }{2}$ and $2\pi$ to obtain the key points as follows:

Key points	$x$	$f\left(x\right)=\sin x$	$\left(x,y\right)$
Intercept	0	0	$\left(0,0\right)$
Maximum	$\frac{\pi }{2}$	1	$\left(\frac{\pi }{2},1\right)$
Intercept	$\pi$	0	$\left(\pi ,0\right)$
Minimum	$\frac{3\pi }{2}$	$-1$	$\left(\frac{3\pi }{2},-1\right)$
Intercept	$2\pi$	0	$\left(2\pi ,0\right)$

The amplitude of the graph $g\left(x\right)$ is 4 times more than amplitude of the graph $f\left(x\right)$ . Draw the graph of the functions $f\left(x\right)$ and $g\left(x\right)$ on a same rectangular coordinate system as follows: