A graph is symmetric with respect to the x-axis if whenever a point $\left(x\,y\right)$ is on the graph the point $\left(x\,-y\right)$ is also on the graph.

The following graph is symmetric with respect to the $\mathrm{x}$-axis. The mirror image of the blue part of the graph in the $\mathrm{x}$-axis is just the red part, and vice versa.

This graph is that of the curve $x \= y^2-1$. If you replace $y$ with $-y$, the result is $x \= {\left(-y\right)}^2-1 \= y^2-1$, which mathematically shows that this graph is symmetric about the x-axis.

A graph is symmetric with respect to the y-axis if whenever a point $\left(x\,y\right)$ is on the graph the point $\left(-x\,y\right)$ is also on the graph.

This graph is symmetric with respect to the $y$-axis. The mirror image of the blue part of the graph in the y-axis is just the red part, and vice versa.

This graph is that of the curve $y\=x^4-3 x^2\+1$. If you replace $x$ with $-x$ the result is $y\={\left(-x\right)}^4-3{\left(-x\right)}^2\+1\=x^4-3 x^2\+1$, which mathematically shows that this graph is symmetric about the y-axis.

A graph is symmetric with respect to the origin if whenever a point $\left(x\,y\right)$ is on the graph the point $\left(-x\,-y\right)$ is also on the graph.

This graph is symmetric with respect to the origin.

This is the graph of the curve $y\=x^3-2 x$. If you replace $x$ with$-x$ and $y$ with $-y$ the result is $-y\={\left(-x\right)}^3-2 \left(-x\right)\=-\left(x^3-2 x\right)$, which on multiplication of both sides by $-1$ gives $y\=x^3-2 x$, the original equation. This mathematically shows that this graph is symmetric with respect to the origin.