$y=x^3$
Step 1:
(i) Domain, extent, intercepts and orgin
The function is defined for all real values of x and hence the domain is the entire interval $(-\infty, \infty)$.
The horizontal extent is $-\infty < x < \infty $ and the vertical extent is $-\infty < y <\infty$ and the vertical extent is $ -\infty < y < \infty$
The curve passes through the orgin.
Since when$ x=0,y=0$. The curve does not intersect the coordinate axes anywhere else
Step 2:
(ii) Symmetry Test : The test for symmetry shows that the curve is symmetric about the origin.
The equation does not change when x is replaced by -x by -y simultaneously
Step 3:
(iii) Asymptotes: There are no constant c,k for which $ y \to \pm \infty$ when $ x \to c$ or $ x \to \pm \infty $ then $ y \to k$
Therefore, the curve does not have any asymptotes
Step 4:
Monotonicity: The first derivative test shows that the curve is increasing throughout $(-\infty, \infty)$
Step 5:
Special points : The curve is concave downward in $(-\infty,0)$ and concave upward in $(0, \infty)$. Since $y^x=6 x < 0$ for $ x < 0 $ and $y''=6x > 0$ for $ x > 0$ and $y''=0$ for $x=0,$ yielding $(0,0)$ as a point of inflection