Let's start with the original parabola given by the equation:
\[ y = x^2 \]
Step 1: Reflect Across the \(x\)-Axis
Reflecting this parabola across the \(x\)-axis changes the sign of \(y\). Therefore, the equation becomes:
\[ y = -x^2 \]
Step 2: Scale Vertically by a Factor of \(\frac{1}{6}\)
Now, we scale the new equation vertically by a factor of \(\frac{1}{6}\). This involves multiplying the entire equation by \(\frac{1}{6}\):
\[ y = \frac{1}{6}(-x^2) = -\frac{1}{6}x^2 \]
Final Equation
Thus, the equation of the new parabola after reflection across the \(x\)-axis and vertical scaling by a factor of \(\frac{1}{6}\) is:
\[ \boxed{-\frac{1}{6}x^2} \]