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