Answer
The function is symmetric about the origin.
Work Step by Step
$f(x)=x|x|$
Check for symmetry about the $y$-axis by substituting $x$ by $-x$ in the given function and simplifying:
$f(-x)=-x|-x|=-x|x|$
Since substituting $x$ by $-x$ did not yield an equivalent expression, the function is not symmetric about the $y$-axis.
Check for symmetry about the $x$-axis by substituting $f(x)$ by $-f(x)$ in the given function and simplifying:
$-f(x)=x|x|$
Multiply both sides by $-1$:
$f(x)=-x|x|$
Since substituting $f(x)$ by $-f(x)$ did not yield an equivalent expression, the function is not symmetric about the $y$-axis.
Check for symmetry about the origin by substituting $x$ by $-x$ and $f(x)$ by $-f(x)$ and simplifying:
$-f(x)=-x|-x|$
$-f(x)=-x|x|$
Multiply both sides by $-1$:
$f(x)=x|x|$
Since the substitution yielded an equivalent expression, the function is symmetric about the origin.
The graph is shown in the answer section
