Answer
The x-intercepts are (-1,0), (0,0), and (1,0)
The y-intercept is (0,0)
The equation has symmetry only with respect to the origin.
Work Step by Step
To find the x-intercept(s), we set y to 0 and solve for x:
$0=x^3-x$
$0=x(x^2-1)$
$x_1=0$
$0=(x_2)^2-1\rightarrow \sqrt{(x_2)^2}=\sqrt1 \rightarrow x_2=\pm1$
To find the y-intercept(s), we set x to y and solve for y:
$y=0^3-0$
$y=0$
To test for symmetry with respect to the x-axis, we substitute y for -y and check if it equals the original equation:
$(-y)=x^3-x$
$-y=x^3-x$ nope
To test for symmetry with respect to the y-axis, we substitute x for -x and check if it equals the original equation:
$y=(-x)^3-(-x)$
$y=-x^3+x$ nope
To test for symmetry with respect to the origin, we substitute x for -x, substitute y for -y and check if it equals the original equation:
$(-y)=(-x)^3-(-x)$
$-y=-(x^3-x)$
$y=x^3-x\checkmark$