The curve $C$ defined by $y = \frac{ax + 1}{bx - 2}$ passes through $(3, 13)$ and has vertical asymptote $x = 2$.

1. Find the value of $b$ using the condition for the vertical asymptote.

The vertical asymptote is $x = 2$, which occurs when the denominator is zero.
$b(2) - 2 = 0$
$2b = 2$
$b = 1$

2. Find the value of $a$ by substituting the given coordinates of the point on the curve.

Substitute $x = 3$, $y = 13$, and $b = 1$ into the equation $y = \frac{ax + 1}{bx - 2}$:
$13 = \frac{a(3) + 1}{(1)(3) - 2}$
$13 = \frac{3a + 1}{1}$
$13 = 3a + 1$
$3a = 12$
$a = 4$

1. Determine the equation of curve $C$ using the values of $a$ and $b$ from part (i), and express it in proper fraction form.

From part (i), $a = 4$ and $b = 1$.
The equation of curve $C$ is $y = \dfrac{4x + 1}{x - 2}$.
Using algebraic manipulation (or long division) to split the fraction:
$y = \dfrac{4(x - 2) + 9}{x - 2}$
$y = \dfrac{9}{x - 2} + 4$

2. Identify the horizontal translation.

Starting from $y = \dfrac{1}{x}$, replacing $x$ with $x - 2$ gives $y = \dfrac{1}{x - 2}$.
This corresponds to a translation of $2$ units in the positive $x$-direction.

3. Identify the vertical scaling.

Multiplying the function by $9$ gives $y = 9 \left( \dfrac{1}{x - 2} \right) = \dfrac{9}{x - 2}$.
This corresponds to a vertical stretch (scaling parallel to the $y$-axis) by a scale factor of $9$.

4. Identify the vertical translation.

Adding $4$ to the function gives $y = \dfrac{9}{x - 2} + 4$.
This corresponds to a translation of $4$ units in the positive $y$-direction.