Chapter 4.4, Problem 69E

Solution

To calculate: The values of $a,b,h,\text{and}k$ so that the graph of the function $y=a\sin \left[b\left(x+h\right)\right]+k$ is curve shown.

The viewing window of the graph is $\left[0,6.28\right]\text{ by }\left[-4,4\right]$ .

  

The resultant values are $a=2,b=2,h=0,k=0$ .

Given information:

The given graph is:

  

Formula used: The period formula is $\text{Period}=\frac{2\pi }{\left|b\right|}$ .

Calculation:

Consider the given graph:

  

The function is: $y=a\sin \left[b\left(x+h\right)\right]+k$

Here $a$ is the amplitude of the graph and $b$ are constant, h is phase shift and k is vertical shift.

Observe the graph, the maximum value is 2 and minimum value is -2.

So the Amplitude is:

  $\begin{array}{c}\text{Amplitude}\left(a\right)=\frac{M-m}{2}\\ =\frac{2-\left(-2\right)}{2}\\ =2\end{array}$

The vertical shift is:

  $\begin{array}{c}\text{Vertical shift}\left(k\right)=\frac{M+m}{2}\\ =\frac{2+\left(-2\right)}{2}\\ =0\end{array}$

The graph completes a single cycle in $\pi$ units, so the period of the function is $\pi$ .

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\left|b\right|}\\ \pi =\frac{2\pi }{\left|b\right|}\\ \left|b\right|=2\\ b=2\end{array}$

The graph goes through the point $\left(0,0\right)$ . Then $h=0$ .

Then the resultant values are $a=2,b=2,h=0,k=0$ .

The function will become:

  $\begin{array}{l}y=2\sin \left[2\left(x+0\right)\right]+0\\ y=2\sin 2x\end{array}$