To find the equation of the new parabola, we will first reflect the original parabola \(y = x^2\) across the \(x\)-axis, and then scale it vertically by a factor of \(\frac{1}{8}\).
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Reflecting across the \(x\)-axis: Reflecting the parabola \(y = x^2\) across the \(x\)-axis means changing the sign of \(y\): \[ y = -x^2 \]
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Scaling vertically by a factor of \(\frac{1}{8}\): To scale the parabola vertically, we multiply the entire equation by \(\frac{1}{8}\): \[ y = \frac{1}{8}(-x^2) = -\frac{1}{8}x^2 \]
Thus, the equation of the new parabola after reflecting across the \(x\)-axis and scaling by a factor of \(\frac{1}{8}\) is: \[ \boxed{y = -\frac{1}{8}x^2} \]