Answer
$\underline{\text{The equation is:}}$
Not symmetric to the $X-$axis.
Not symmetric to the $Y-$axis.
Symmetric with respect to the origin.
Work Step by Step
$\color{red}{\text{If $(x,y)$ exists on the graph,}}$
$\color{red}{\text{ then the graph is symmetric about the:}}$
1. $X-$Axis if $(x,−y)$ exists on the graph,
2. $Y-$Axis if $(−x,y)$ exists on the graph,
3. Origin if $(−x,−y)$ exists on the graph
$\color{blue}{\text{Test for symmetry about the $X-$axis by plugging in $−y$ for $y$.}}$
$-y=x^3+10x$
Remove parentheses.
$-y=x^3+10x$
Since the equation is not identical to the original equation, it is not symmetric to the $X-$axis.
Not symmetric to the $X-$axis.
$\color{blue}{\text{Test for symmetry about the $Y-$axis by plugging in $−x$ for $x$.}}$
$y=(-x)^3+10(-x)$
Remove parentheses.
$y=-x^3-10x$
Since the equation is not identical to the original equation, it is not symmetric to the $Y-$axis.
Not symmetric to the $Y-$axis.
$\color{blue}{\text{Test for symmetry about the origin axis by plugging in $−x$ for $x$ and $-y$ for $y$.}}$
$-y=(-x)^3+10(-x)$
Remove parentheses.
$y=x^3+10x$
Since the equation is identical to the original equation, it is symmetric with respect to the origin.
Symmetric with respect to the origin.