By expressing the equation of the curve $y = \frac{12x + 11}{2x + 1}$ in the form $y = A + \frac{B}{2x + 1}$, where $A$ and $B$ are constants, describe a sequence of three transformations which maps the graph of $y = \frac{1}{2x - 3}$ onto the graph of $y = \frac{12x + 11}{2x + 1}$. The diagram shows the graph of $y = \mathrm{f}(x)$. The curve has a maximum point at $(0, -5)$ and a minimum point at $(6, -4)$. The equations of the asymptotes of the curve are $x = -2$, $x = 2$ and $y = -2$. Sketch the graph of $y = \mathrm{f}(-x + 2) + 4$, indicating clearly the equations of the asymptotes and the coordinates of the axial intercepts and turning points.

1. Express the given equation in the required partial fraction form to identify constants $A$ and $B$.

$y = \frac{12x + 11}{2x + 1}$
$y = \frac{6(2x + 1) + 5}{2x + 1}$
$y = 6 + \frac{5}{2x + 1}$
Thus, $A = 6$ and $B = 5$.

2. Determine the horizontal translation required to transform the denominator.

We start with the original curve $\mathrm{f}(x) = \frac{1}{2x - 3}$.
To change the denominator to $2x + 1$, we need to replace $x$ with $x + 2$:
$\mathrm{f}(x + 2) = \frac{1}{2(x + 2) - 3} = \frac{1}{2x + 1}$
This corresponds to a translation of $-2$ units parallel to the $x$-axis.

3. Determine the vertical scaling required to match the numerator.

Next, we multiply the function by $5$ to get the required numerator of $B = 5$:
$5\mathrm{f}(x + 2) = \frac{5}{2x + 1}$
This corresponds to a scaling parallel to the $y$-axis with a scale factor of $5$.

4. Determine the vertical translation required to complete the function mapping.

Finally, we add $6$ to the function to account for $A = 6$:
$5\mathrm{f}(x + 2) + 6 = 6 + \frac{5}{2x + 1}$
This corresponds to a translation of $6$ units parallel to the $y$-axis.