Question

True or False: The equation $y^2=9+x^2$is symmetric with

respect to the $x-$axis, the $y-$axis, and the origin.

Short answer

Yes, the equation $y^2=9+x^2$is symmetric with respect to the $x-$axis,$y-$axis and the origin.

Step-by-step solution

Step 1. Given Information

The given equation is$y^2=9+x^2$.

Step 2. Objective of the Problem

We need to find if the equation $y^2=9+x^2$is symmetric with respect to the $x-$axis, $y-$axis and the origin.

Step 3. Concept

A graph is said to be symmetric with respect to the $x-$axis if, for every point

$(x,y)$ on the graph, the point $(x,-y)$ is also on the graph.

A graph is said to be symmetric with respect to the $y-$axis if, for every point

$(x,y)$ on the graph, the point $(-x,y)$ is also on the graph.

A graph is said to be symmetric with respect to the origin if, for every point

$(x,y)$ on the graph, the point $(-x,-y)$ is also on the graph.

Symmetric with respect to the x-axis.

To test the graph of an equation for symmetry with respect to the $x-$Axis we must replace $y$by $-y$in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the $x-$axis.

In the equation $y^2=9+x^2$, replace $y$by $-y.$

$$\begin{gathered} {(-y)}^2=9+x^2 \\ {y}^2=9+x^2 \end{gathered}$$

Since we get back the original equation, we say that the equation $y^2=9+x^2$is symmetric with respect to the $x-$axis.

Symmetric with respect to the y-axis.

To test the graph of an equation for symmetry with respect to the $y-$Axis we must replace $x$by $-x$in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the $y-$axis.

In the equation $y^2=9+x^2$, replace $x$by $-x.$

$y^2=9+{(-x)}^2{y}^2=9+x^2$

Since we get back the original equation, we say that the equation $y^2=9+x^2$is symmetric with respect to the $y-$axis.

Step 5. Symmetric with respect to the origin.

To test the graph of an equation for symmetry with respect to the origin we must replace $y$by $-y$ and $x$in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin.

In the equation $y^2=9+x^2$, replace $y$by $-y$and $x$by$-x.$

role="math" localid="1647285938546" $$\begin{gathered} {(-y)}^2=9+{(-x)}^2 \\ {y}^2=9+x^2 \end{gathered}$$

Since we get back the original equation, we say that the equation $y^2=9+x^2$is symmetric with respect to the origin.