Answer
1) The $x$-intercept is $(-4,0)$, and the $y$-intercepts are $(0,2)$ and $(0,-2)$
2) The graph of the equation ${{y}^{2}}=x+4$ is symmetric with respect to the $x$-axis.
3) The graph of the equation ${{y}^{2}}=x+4$ is not symmetric with respect to the $y$-axis, the $x$-axis, or the origin.
Work Step by Step
To find $x$-intercept(s), let $y=0$
$\Rightarrow \,0=x+4$
$\Rightarrow \,\,x=-4$.
Therefore, the $x$-intercept is $(-4,0)$
To find the $y$-intercept(s), let $x=0$
$\Rightarrow \,\,{{y}^{2}}=0+4$
$\Rightarrow \,{{y}^{2}}=4$
$\Rightarrow \,\,y=\pm \,2$.
The $y$-intercepts are $(0,2)$ and $(0,-2)$
To test for symmetry with respect to the $x$-axis, replace $y$ with $-y$. If it results in an expression equivalent to the original equation, then the equation is symmetric with respect to the $x$-axis.
$\Rightarrow $${{(-y)}^{2}}=x+4$ is equivalent to ${{y}^{2}}=x+4$
Hence, the graph of the equation ${{y}^{2}}=x+4$ is symmetric with respect to the $x$-axis.
To test for symmetry with respect to the $y$-axis, replace $x$ with –$x$. If it results in an expression equivalent to the original equation, then the equation is symmetric with respect to the $y$-axis.
$\Rightarrow \,{{y}^{2}}=-x+4$ is not equivalent to ${{y}^{2}}=x+4$
Hence, the graph of the equation ${{y}^{2}}=x+4$ is not symmetric with respect to the $y$-axis.
To test for symmetry with respect to the origin, replace $x$ with –$x$ and $y$ with $-y$. If it results in an expression equivalent to the original equation, then the equation is symmetric with respect to the origin.
$\Rightarrow \,\,{{(-y)}^{2}}=-x+4$
$\Rightarrow \,{{y}^{2}}=-x+4$ is not equivalent to ${{y}^{2}}=x+4$.
Hence, the graph of the equation ${{y}^{2}}=x+4$ is not symmetric with respect to the origin.
