Precalculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 16

Find the intercepts of the equation $x^2 + 4x + y^2 - 2y = 0$ and test for symmetry with respect to the $x$-axis, the $y$-axis and the origin.

$x$-intercepts


$\begin{equation} \begin{aligned} & x^2 + 4x + y^2 - 2y = 0 && \text{Given equation} \\ & x^2 + 4x + (0)^2 - 2(0) = 0 && \text{To find the $x$-intercept, we let $y = 0$ and solve for $x$} \\ & x^2 + 4x = 0 && \\ & x (x + 4) = 0 && \\ & x = 0 \text{ and } x + 4 = 0 && \\ & x = 0 \text{ and } x = -4 && \end{aligned} \end{equation}$


The $x$-intercepts are $(0,0)$ and $(-4,0)$

$y$-intercepts


$\begin{equation} \begin{aligned} & x^2 + 4x + y^2 - 2y = 0 && \text{Given equation} \\ & (0)^2 + 4(0) + y^2 - 2y = 0 && \text{To find the $y$-intercept, we let $x = 0$ and solve for $y$} \\ & y^2 - 2y = 0 && \\ & y(y-2) = 0 && \\ & y = 0 \text{ and } y - 2 = 0 && \\ & y = 0 \text{ and } y = 2 \end{aligned} \end{equation}$


The $y$-intercepts are $(0,0)$ and $(0,2)$.

Test for symmetry

$x$-axis


$\begin{equation} \begin{aligned} x^2 + 4x + y^2 - 2y =& 0 && \text{Given equation} \\ x^2 + 4x + (-y)^2 - 2(-y) =& 0 && \text{To test for $x$-axis symmetry, replace $y$ by $-y$ and see if the equation is still the same} \\ x^2 + 4x + y^2 + 2y =& 0 && \end{aligned} \end{equation}$


The equation changes so the equation is not symmetric to $x$-axis.

$y$-axis


$\begin{equation} \begin{aligned} x^2 + 4x + y^2 - 2y =& 0 && \text{Given equation} \\ (-x)^2 + 4(-x) + y^2 - 2y =& 0 && \text{To test for $y$-axis symmetry, replace$ x$ by $-x$ and see if the equation is still the same} \\ x^2 - 4x + y^2 - 2y =& 0 && \end{aligned} \end{equation}$


The equation changes so the equation is not symmetric to $y$-axis.

Origin


$\begin{equation} \begin{aligned} & x^2 + 4x + y^2 - 2y = 0 && \text{Given equation} \\ & (-x)^2 + 4(-x) + (-y)^2 - 2(-y) = 0 && \text{To test for origin symmetry, replace both $x$ by $-x$ and y by $-y$ and see if the equation is still the same} \\ & x^2 - 4x + y^2 + 2y = 0 && \end{aligned} \end{equation}$


The equation changes so the equation is not symmetric to the origin.