Chapter 5.1, Problem 83E

Solution

To plot: The graph of the functions $y={\sin }^{2}x$ and $y=-{\cos }^{2}x$ , and also describe the relationship between two graphs.

The graph of the functions $y={\sin }^{2}x$ and $y=-{\cos }^{2}x$ is shown in figure (1), and graph of function $y=-{\cos }^{2}x$ is obtained by shifting the graph of function $y={\sin }^{2}x$ by $1$ unit down.

Given information:

The functions are $y={\sin }^{2}x$ and $y=-{\cos }^{2}x$ .

Formula used:

The Pythagorean identity is given by,

  ${\sin }^{2}x+{\cos }^{2}x=1$

Calculation:

Plot the graph of the functions $y={\sin }^{2}x$ and $y=-{\cos }^{2}x$ as follows.

  

Figure (1)

From the graph, it can be observed that the graph of the functions $y={\sin }^{2}x$ and $y=-{\cos }^{2}x$ are same but the graph of $y=-{\cos }^{2}x$ is translated $1$ unit down.

From the Pythagorean identity,

  $\begin{array}{c}{\sin }^{2}x+{\cos }^{2}x=1\\ -{\cos }^{2}x={\sin }^{2}x-1\end{array}$

It is known that in the formula $y=f\left(x\right)+c$ , $c$ represents shift of the graph along $y$ axis.

Therefore, the graph of function $y=-{\cos }^{2}x$ is obtained by shifting the graph of function $y={\sin }^{2}x$ by $1$ unit down.