Answer
$x=-3,0,3$, $y=0$.
symmetric with respect to the origin.
Work Step by Step
Step 1. Given $y=x^3-9x=x(x+3)(x-3)$, we can find the x-intercepts (zeros) $x=-3,0,3$, y-intercept(s) $(0,0)$.
Step 2. Test symmetry with respect to the x-axis, replace $(x,y)$ with $(x,-y)$, we have $-y=x^3-9x$ and it is not the same as the original meaning it is not symmetric with respect to the x-axis.
Step 3. Test symmetry with respect to the y-axis, replace $(x,y)$ with $(-x,y)$, we have $y=-x^3+9x$ and it is not the same as the original meaning it is not symmetric with respect to the y-axis.
Step 4. Test symmetry with respect to the origin, replace $(x,y)$ with $(-x,-y)$, we have $-y=-x^3+9x \Longrightarrow y=x^3-9x$ and it is the same as the original meaning it is symmetric with respect to the origin.
