Chapter 13.4, Problem 13WE

To determine

The curve and mention its amplitude and period.

Answer to Problem 13WE

The given curve is sine curve, amplitude is 2 and the period is 2π .

Explanation of Solution

Given:

  

Calculations:

Suppose general form of a function,

  $y=c+a\sin bx$ ,

Here, a is the amplitude of the function.

And period is $\frac{2\pi }{b}$ .

Also,

The average of sum of maximum and minimum gives,

  $c=\frac{M+m}{2}$

The average of difference of maximum and minimum value gives amplitude,

  $a=\frac{M-m}{2}$

From the graph,

The value of curve at origin is 0.

  $\sin 0=0$

The given curve is a sine curve. Since, the value of sine 0 is 0.

The curve completes a cycle in interval of 2π . So, the period of the given cycle is 2π.

The maximum value of the curve is 2 and the minimum value is -2.

Therefore,

  $\begin{array}{l}\frac{2\pi }{b}=2\pi \\ b=\frac{2\pi }{2\pi }\\ b=1\end{array}$

We know that,

The maximum value of the curve is 2 and the minimum value is -2.

Therefore,

  $\begin{array}{l}c=\frac{M+m}{2}=\frac{2+\left(-2\right)}{2}=0\\ a=\frac{M-m}{2}=\frac{2-\left(-2\right)}{2}=2\end{array}$

So, the equation of the graph is

  $\begin{array}{l}y=0+\left(2\right)\sin \left(1\right)x\\ y=2\sin x\end{array}$