$y=x^3$

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