Chapter 1.8, Problem 67E

To determine

To calculate: The x- and y- intercepts for the equation $xy=2$ and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Answer to Problem 67E

There are nox-intercepts and y-intercepts. The equation is symmetric about origin. Graph of the equation is provided below,

  

Explanation of Solution

Given information:

The equation $xy=2$ .

Formula used:

The function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Calculation:

It is provided that the equation is $xy=2$ . Construct a table to evaluate the value of y for different values of x.

Substitute the point $x=1$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(1\right)y=2\\ y=2\end{array}$

Substitute the point $x=2$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(2\right)y=2\\ y=1\end{array}$

Substitute the point $x=3$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(3\right)y=2\\ y=\frac{2}{3}\end{array}$

Substitute the point $x=4$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(4\right)y=2\\ y=\frac{1}{2}\end{array}$

Construct a table with the values obtained above,

  $\begin{array}{ccc}x& y& \left(x,y\right)\\ 1& 2& \left(1,2\right)\\ 2& 1& \left(2,1\right)\\ 3& \frac{2}{3}& \left(3,\frac{2}{3}\right)\\ 4& \frac{1}{2}& \left(4,\frac{1}{2}\right)\end{array}$

In the coordinate plane plot the points obtained above and connect them through a line.

The graph of the equation is provided below $xy=2$ .

  

Recall that the x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

Substitute $y=0$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(0\right)x=2\\ x=\frac{2}{0}\end{array}$

Therefore, there are no x-intercepts.

Recall that the y-intercepts are the points on x-axis where the graph of the equation intersects the y-axis.

Substitute $x=0$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(0\right)y=2\\ y=\frac{2}{0}\end{array}$

Therefore, there are noy-intercepts.

Recall that the function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

Replace y by $-y$ in the equation $xy=2$ ,

  $\begin{array}{c}x\left(-y\right)=2\\ -xy=2\end{array}$

The equation is changed. Therefore, the equation $y=9-x^2$ is not symmetricabout the x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(-x\right)y=2\\ -xy=2\end{array}$

The equation is changed. Therefore, the equation $xy=2$ is not symmetricabout the y-axis.

Recall that the function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ and y by $-y$ in the equation $xy=2$ ,

  $\begin{array}{c}\left(-x\right)\left(-y\right)=2\\ xy=2\end{array}$

The equation is unchanged. Therefore, the equation $xy=2$ is symmetricabout the origin.

Thus, there are nox-intercepts and y-intercepts. The equation is symmetric about origin.Graph of the equation $xy=2$ is provided below,