Question

The parabola
\[y=x^2\] is reflected across the
\[x\]-axis and then scaled vertically by a factor of

\[\dfrac{1}8\].
What is the equation of the new parabola?
\[y=\]

Answer 1

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 \(\frac{1}{8}\).

1. **Reflecting across the \(x\)-axis:**
Reflecting the parabola \(y = x^2\) across the \(x\)-axis means changing the sign of \(y\):
\[
y = -x^2
\]

2. **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:
\[
\boxed{y = -\frac{1}{8}x^2}
\]