曲线 $C$ 依次进行以下变换： $A$：沿 $y$ 轴负方向平移 4 个单位。 $B$：沿平行于 $x$ 轴的方向拉伸至原来的 2 倍。 $C$：关于 $x$ 轴反射。 变换后得到的曲线方程为 $y = \frac{x^2}{4} + 4$。 求变换前的原始曲线 $C$ 的方程。

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 $y = -x^2$.