A curve $C$ undergoes, in succession, the following transformations. $A$: Translation of 4 units in the negative $y$-direction. $B$: Stretching parallel to the $x$-axis by a factor of 2. $C$: Reflection in the $x$-axis. The equation of the resulting curve is $y = \frac{x^2}{4} + 4$. Determine the equation of the original curve $C$ before the transformations were carried out.

1. Express the final transformed curve in terms of the unknown original function $\mathrm{f}(x)$.

Let the original curve $C$ be $y = \mathrm{f}(x)$.
Applying transformation $A$ (translation of 4 units in the negative $y$-direction):
$y = \mathrm{f}(x) - 4$
Applying transformation $B$ (stretching parallel to the $x$-axis by a factor of 2):
$y = \mathrm{f}\left(\frac{x}{2}\right) - 4$
Applying transformation $C$ (reflection in the $x$-axis):
$y = -\left[\mathrm{f}\left(\frac{x}{2}\right) - 4\right]$
$y = -\mathrm{f}\left(\frac{x}{2}\right) + 4$

2. Equate the transformed expression to the given resulting equation to solve for the original function.

We are given that the resulting curve is $y = \frac{x^2}{4} + 4$.
Equating the two expressions for $y$:
$-\mathrm{f}\left(\frac{x}{2}\right) + 4 = \frac{x^2}{4} + 4$
Subtracting 4 from both sides:
$-\mathrm{f}\left(\frac{x}{2}\right) = \frac{x^2}{4}$
$\mathrm{f}\left(\frac{x}{2}\right) = -\frac{x^2}{4}$
Rewrite the right-hand side in terms of $\frac{x}{2}$:
$\mathrm{f}\left(\frac{x}{2}\right) = -\left(\frac{x}{2}\right)^2$
Substituting $\frac{x}{2}$ with $x$ yields the original function:
$\mathrm{f}(x) = -x^2$
Thus, the equation of the original curve $C$ is $y = -x^2$.