Math Snap

STEP 1

Assumptions1. The equations provided are in the form of $y=f(x)$ or $x=f(y)$ or $f(x,y)=0$.
. We are to determine the symmetry of the graph of each equation with respect to the $y$-axis, the $x$-axis, the origin, more than one of these, or none of these.

STEP 2

To check for symmetry about the $y$-axis, we replace $x$ with $-x$ in the equation and see if we get the same equation back.

STEP 3

To check for symmetry about the $x$-axis, we replace $y$ with $-y$ in the equation and see if we get the same equation back.

STEP 4

To check for symmetry about the origin, we replace $x$ with $-x$ and $y$ with $-y$ in the equation and see if we get the same equation back.

STEP 5

Let's start with the first equation, $y=x^{2}+$. Replace $x$ with $-x$.
$$y=(-x)^{2}+$$

STEP 6

implify the equation.
$$y=x^{2}+6$$

STEP 7

Since we got the same equation back, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$-axis.

STEP 8

Now, let's check for symmetry about the $x$-axis. Replace $y$ with $-y$ in the equation $y=x^{2}+6$.
$$-y=x^{2}+6$$

STEP 9

This is not the same as the original equation, so the graph of the equation $y=x^{2}+6$ is not symmetric with respect to the $x$-axis.

STEP 10

Finally, let's check for symmetry about the origin. Replace $x$ with $-x$ and $y$ with $-y$ in the equation $y=x^{2}+6$.
$$-y=(-x)^{2}+6$$

STEP 11

This is not the same as the original equation, so the graph of the equation $y=x^{}+6$ is not symmetric with respect to the origin.

STEP 12

Therefore, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$-axis only.

Repeat steps5 to12 for each of the remaining equations to determine their symmetry.