Answer
The x-intercepts are (-3,0), (0,0), and (3,0).
The only 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:
$0=x^3-9x$
$0=x(x^2-9)$
$0=x(x+3)(x-3)$
$x_1=-3$
$x_2=0$
$x_3=3$
To find the y-intercept(s), we set x to 0 and solve for y:
$y=0^3-9(0)$
$y=0-0$
$y=0$
To test for symmetry with respect to the x-axis, we substitute y for -y and check if it equals the original equation:
$(-y)=x^3-9x$
$-y=x^3-9x$ nope
To test for symmetry with respect to the y-axis, we substitute x for -x and check if it equals the original equation:
$y=(-x)^3-9(-x)$
$y=-x^3+9x$ 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:
$(-y)=(-x)^3-9(-x)$
$-y=-x^3+9x$
$-y=-(x^3-9x)$
$y=x^3-9x \checkmark$
