Chapter 10.5, Problem 5E

Solution

To write: The function for the given sinusoid graph.

The function for the given sinusoid graph is $y=2\sin \frac{\pi }{2}\left(x-1\right)+4$ .

Given information: The graph of the sinusoid is shown below.

  

Figure (1)

Formula used: The general equation of the sinusoid function is given as:

  $y=a\sin b\left(x-h\right)+k$ or $y=a\cos b\left(x-h\right)+k$

Here, $a$ is amplitude, $h$ is the horizontal shift, $k$ is the vertical shift and $\frac{2\pi }{b}$ is the period.

Calculation:

From the figure (1), the maximum value is $M=6$ and the minimum value is $m=2$ .

Calculated the vertical shift of the graph by the mean of the maximum and minimum values.

  $\begin{array}{c}k=\frac{M+m}{2}\\ =\frac{6+2}{2}\\ =\frac{8}{2}\\ =4\end{array}$

The amplitude can be calculated by the half of the difference between maximum and minimum.

  $\begin{array}{c}\left|a\right|=\frac{M-m}{2}\\ =\frac{6-2}{2}\\ =\frac{4}{2}\\ =2\end{array}$

From the graph it can be observed that the period is $4$ . Calculate the value of $b$ .

  $\begin{array}{c}P=\frac{2\pi }{b}\\ 4=\frac{2\pi }{b}\\ b=\frac{\pi }{2}\end{array}$

Substitute $2$ for $a$ , $4$ for $k$ and $\frac{\pi }{2}$ for $b$ in the general equation of sine.

  $y=2\sin \frac{\pi }{2}\left(x-h\right)+4$ …… (1)

To find the horizontal shift, substitute $0$ for $x$ and $2$ for $y$ in the above equation.

  $\begin{array}{c}2=2\sin \frac{\pi }{2}\left(0-h\right)+4\\ -2=2\sin \frac{\pi }{2}\left(-h\right)\\ -1=-\sin \frac{\pi }{2}h\\ \sin \frac{\pi }{2}h=\sin \frac{\pi }{2}\end{array}$

Further simplify the above equation.

  $\begin{array}{c}\frac{\pi }{2}h=\frac{\pi }{2}\\ h=1\end{array}$

Substitute $1$ for $h$ in the equation (1).

  $y=2\sin \frac{\pi }{2}\left(x-1\right)+4$

Therefore, the function for the given sinusoid graph is $y=2\sin \frac{\pi }{2}\left(x-1\right)+4$ .