Chapter 1, Problem 16RE

To determine

To list: the intercepts and to test for symmetry

Answer to Problem 16RE

The $x$ intercepts are 0 and -4.

The $y$ intercepts are 0 and 2.

The graph ofthe equation is not symmetric with respect to the $x$ -axis, $y$ -axis and origin.

Explanation of Solution

Given:

  $x^2+4x+y^2-2y=0$

Calculation:

In order to find the $x$ -intercept, substitute 0 for $y$ in the given equation and evaluate.

  $\begin{array}{r}\hfill x^2+4x+0^2-2(0)=0\\ \hfill x^2+4x=0\end{array}$

Factor the left side.

  $x(x+4)=0$

Apply the zero-product property.

  $x=0$ or $x+4=0$

Solve for $x$

  $x=0$ or $x=-4$

The $x$ intercepts are 0 and -4

Now, substitute 0 for $x$ and evaluate to find the $y$ -intercept.

  $\begin{array}{r}\hfill 0^2+4(0)+y^2-2y=0\\ \hfill y^2-2y=0\end{array}$

Factor the left side.

  $y(y-2)=0$

Apply the zero-product property.

  $y=0\text{ or }y-2=0$

Solve for $y$

  $y=0$ or $y=2$

The $y$ intercepts are 0 and 2.

We can test for the symmetry.

  $x-Axis$ :

In order to test for symmetry with respect to the $x$ -axis, replace $y$ by $-y$ .

  $\begin{array}{r}\hfill x^2+4x+{(-y)}^2-2(-y)=0\\ \hfill x^2+4x+y^2+2y=0\end{array}$

Since $x^2+4x+y^2+2y=0$ is not equivalent to $x^2+4x+y^2-2y=0$ , so the graph ofthe equation is not symmetric with respect to the $x$ -axis.

  $y-Axis:$

Replace $x$ by $-x,$ to test for symmetry with respect to the $y$ -axis

  $\begin{array}{r}\hfill {(-x)}^2+4(-x)+y^2-2y=0\\ \hfill x^2-4x+y^2-2y=0\end{array}$

Since $x^2-4x+y^2-2y=0$ is not equivalent to $x^2+4x+y^2-2y=0$ , the graph of the equation is not symmetric with respect to the $y$ -axis.

Origin:

To test for symmetry with respect to the origin, replace $x$ by $-x,$ and $y$ by $-y$ .

  $\begin{array}{r}\hfill {(-x)}^2+4(-x)+{(-y)}^2-2(-y)=0\\ \hfill x^2-4x+y^2+2y=0\end{array}$

Since $x^2-4x+y^2+2y=0$ is not equivalent to $x^2+4x+y^2-2y=0$ . the graph of the equation is not symmetric with respect to the origin.

Conclusion:

The $x$ intercepts are 0 and -4.

The $y$ intercepts are 0 and 2.

The graph of the equation is not symmetric with respect to the $x$ -axis, $y$ -axis and origin.