Quick questions on Graphs of functions and curve sketching explained: O-Level E-Maths

The basic cubic $y = x^3$ passes through the origin, increases for all $x$, and has rotational symmetry about the origin, falling steeply on the left and rising steeply on the right. Cubics of the form $y = ax^3$ keep this shape, steeper for larger $|a|$ and reflected when $a$ is negative.

The reciprocal $y = \dfrac{1}{x}$ has two separate branches, one in the top-right and one in the bottom-left for $y = \dfrac{1}{x}$. The curve never touches either axis: the $x$-axis and $y$-axis are asymptotes, lines the curve approaches but never meets. There is no value at $x = 0$.

The exponential $y = a^x$ (with $a > 1$) passes through $(0, 1)$, since any base to the power zero is $1$, and increases ever more steeply as $x$ grows. As $x$ becomes large and negative, the curve approaches the $x$-axis from above without touching it, so $y = 0$ is an asymptote and $y$ is always positive.

State the coordinates of the point where $y = 3^x$ crosses the $y$-axis. [1 mark]

State the equations of the two asymptotes of $y = \dfrac{1}{x}$. [2 marks]

Describe one way the graph of $y = -x^3$ differs from $y = x^3$. [1 mark]