Answer
There is no symmetry about the x-axis, y-axis and the origin.
See graph
The intercepts: $(-\sqrt[3]{3},0)$, $(0,3)$.
Work Step by Step
$$y=x^3+3$$
Testing for y-symmetry:
$$y=(-x)^3+3$$ $$y=-x^3+2x$$
Since the resulting equation is not the same as the original equation, there is no y-symmetry.
Testing for x-symmetry:
$$-y=x^3+3$$ $$y=-x^3-3$$
Since the resulting equation is not the same as the original equation, there is no x-symmetry.
Testing for origin-symmetry:
$$-y=(-x)^3+3$$ $$-y=-x^3+3$$ $$y=x^3-3$$
Since the resulting equation is not the same as the original equation, there is no origin-symmetry.
Finding the x-intercepts where $y=0$:
$$0=x^3+3$$
$$x^3=-3$$
$$x=\sqrt[3]{-3}$$
$$x=-\sqrt[3]{3}$$
Thus, the x-intercept is $(-\sqrt[3]{3},0)$.
Finding the y-intercept where $x=0$:
$$y=0^3+3$$
$$y=3$$
Thus, the y-intercept is $(0,3)$.
Finding another point by taking $x=-2$:
$$y=(-2)^3+3=-5$$
Thus, another point is at $(-2,-5)$.
Using the points, the graph is as shown.
The intercepts are for x-intercept is $(-\sqrt[3]{3},0)$ and for y-intercept is $(0,3)$.
