Chapter 9.4, Problem 36E

To determine

To find: The error and find the correct answer for determining the point where the maximum value of the function $y=2\sin \left(x-\frac{\pi }{2}\right)$ .

Answer to Problem 36E

The formula for maximum is wrong is the correct formula is $\left(\frac{1}{4}\times \frac{2\pi }{b}+h,a\right)$ and the correct answer is $\left(\pi ,2\right)$ .

Explanation of Solution

Given:

The given function is $y=2\sin \left(x-\frac{\pi }{2}\right)$ .

The given method to determine the value of points is,

  $\left(\left(\frac{1}{4},2\pi \right)-\frac{\pi }{2},2\right)=\left(\frac{\pi }{2}\frac{\pi }{2},2\right)=\left(0,2\right)$

Calculation:

Consider the given function is,

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

The given method to determine the value of points is,

  $\left(\left(\frac{1}{4},2\pi \right)-\frac{\pi }{2},2\right)=\left(\frac{\pi }{2}\frac{\pi }{2},2\right)=\left(0,2\right)$

Consider the general form the function is,

  $y=a\sin b\left(x-h\right)+k$

Consider the horizontal shift is $\frac{\pi }{2}$ and the vertical shift is

  $k=0$ .

Then, the maximum point is obtained as,

  $\begin{array}{c}\left(\frac{1}{4}\times \frac{2\pi }{b}+h,a\right)=\left(\frac{1}{4}\times \frac{2\pi }{1}+\frac{\pi }{2},2\right)\\ =\left(\frac{\pi }{2}+\frac{\pi }{2},2\right)\\ =\left(\pi ,2\right)\end{array}$

Thus, the formula for maximum is wrong is the correct formula is $\left(\frac{1}{4}\times \frac{2\pi }{b}+h,a\right)$ and the correct answer is $\left(\pi ,2\right)$ .