Chapter 1, Problem 15RE

To determine

To list: the intercepts and to test for symmetry

Answer to Problem 15RE

The $x$ -intercepts are (0.0) and (-1,0) and the $y$ -intercepts are (0,0) and (0,-2)

The graph is not symmetric with the $x$ -axis, $y$ -axis or the origin.

Explanation of Solution

Given:

  $x^2+x+y^2+2y=0$

Calculation:

In order to find the $x$ -intercept, let $y=0$ and solve for $x$ .

  $x^2+x=0$

Factor the equation.

  $x^2+x=x(x+1)=0$

Use the zero product property.

  $x=0$

  $x+1=0$

  $x=-1$

There are two solutions for $x$ .

Therefore, the $x$ -intercepts are (0.0) and (-1,0)

For finding the $y$ -intercept, let $x=0$ and solve for $y$

  $y^2+2y=0$

Factor the equation.

  $y^2+2y=y(y+2)=0$

Use the zero product property.

  $y=0$

  $y+2=0$

  $y=-2$

There are two solutions for $y$ .

Therefore, the $y$ -intercepts are (0,0) and (0,-2)

Test for symmetry with respect to the $x$ -axis, $y$ -axis and the origin.

  $x-$ Axis:

For testing symmetry with respect to the $x$ -axis, replace $y$ by $-y$

  $\begin{array}{r}\hfill x^2+x+{(-y)}^2+2(-y)=0\\ \hfill x^2+x+y^2-2y=0\end{array}$

Since this equation is not equivalent to the original equation, the graph of the equation $x^2+x+$

  $y^2+2y=0$ is not symmetric with respect to the $x$ -axis

  $y-$ Axis:

For testing symmetry with respect to the $y$ -axis, replace $x$ by $-x$

  $\begin{array}{r}\hfill {(-x)}^2+(-x)+y^2+2y=0\\ \hfill x^2-x+y^2-2y=0\end{array}$

Since this equation is not equivalent to the original equation, the graph of the equation $x^2+x+$

  $y^2+2y=0$ is not symmetric with respect to the $y$ -axis

Origin:

For testing symmetry with respect to the origin, replace $x$ by $-x$ and

  $y$ by $-y$

  $\begin{array}{r}\hfill {(-x)}^2+(-x)+{(-y)}^2+2(-y)=0\\ \hfill x^2-x+y^2-2y=0\end{array}$

Since this equation is not equivalent to the original equation, the graph of the equation $x^2+x+$

  $y^2+2y=0$ is not symmetric with respect to the origin.

Therefore the intercepts are (0,0),(-1,0) and (0,-2) and the graph is not symmetric with the

  $x$ -axis, $y$ -axis or the origin.

Conclusion:

The $x$ -intercepts are (0.0) and (-1,0) and the $y$ -intercepts are (0,0) and (0,-2)

The graph is not symmetric with the $x$ -axis, $y$ -axis or the origin.