Chapter 4.5, Problem 102E

a.

To determine

How does the value of $c$ affect the graph?

Answer to Problem 102E

  $\boxed{\text{c is horizontal translation of the graph}}$

Explanation of Solution

Given information:

Te figure below shows the graph of $y=\sin \left(\chi -c\right)for$

  $c=\frac{\pi }{4},0,and\frac{\pi }{4}.$

  

  $c=\frac{\pi }{4}$

  $c=0$

  $c=\frac{\pi }{4}$

How does the value of $c$ affect the graph?

Calculation:

As you know that, for $y=\sin \left(\chi -c\right),$ Horizontal shift: $c$ you obtain,

For $y=\sin \left(\chi -\frac{\pi }{4}\right)$ , Horizontal shift: $\frac{\pi }{4}\mathrm{.......}\left(1\right)$

For $y=\sin \chi ,$ Horizontal shift: $\mathrm{0.......}\left(2\right)$

For $y=\sin \left(\chi \frac{\pi }{4}\right),$ Horizontal shift: $-\frac{\pi }{4}\mathrm{.....}\left(3\right)$

Graphs of $y=\sin \left(\chi -\frac{\pi }{4}\right)$ , $y=\sin \chi ,$ and $y=\sin \left(\chi +\frac{\pi }{4}\right)$ are sketched as follows,

From $\left(1\right),\left(2\right),\left(3\right)$ and also from the graph above, you see that as value of $c$ varies, the graph is shifted horizontally. Moreover $\boxed{\text{c is horizontal translation of the graph}}$

b

To determine

Which graph is equivalent to that of $y=-\cos \left(\chi +\frac{\pi }{4}\right)?$

Answer to Problem 102E

  $\boxed{\text{ Graph of y=sin}\left(\chi -\frac{\pi }{4}\right)\text{is equivalent}}$

Explanation of Solution

Given information:

Te figure below shows the graph of $y=\sin \left(\chi -c\right)for$

  $c=\frac{\pi }{4},0,and\frac{\pi }{4}.$

  

  $c=\frac{\pi }{4}$

  $c=0$

  $c=\frac{\pi }{4}$

Which graph is equivalent to that of $y=-\cos \left(\chi +\frac{\pi }{4}\right)?$

Calculation:

For $y=-\cos \left(\chi -\frac{\pi }{4}\right),$

Using the result that co-functions of complementary angles are equal, that is , $\cos \theta =\sin \left(\frac{\pi }{2}-\theta \right),you$ get

  $\begin{array}{l}y=-\sin \left[\frac{\pi }{2}-\left(\chi +\frac{\pi }{4}\right)\right]\\ =-\sin \left[\frac{\pi }{2}-\chi -\frac{\pi }{4}\right]\\ =-\sin \left[-\chi +\frac{\pi }{4}\right]\\ \\ \end{array}$

Using the result, $\sin \left(-\theta \right)=-\sin \theta$

  $\begin{array}{l}y=\sin \left[-\left(-\chi +\frac{\pi }{4}\right)\right]\\ =\sin \left(\chi -\frac{\pi }{4}\right)\\ \end{array}$

Therefore $\boxed{\text{ Graph of y=sin}\left(\chi -\frac{\pi }{4}\right)\text{is equivalent}}$ to that of $y=-\cos \left(\chi +\frac{\pi }{4}\right)$