The parabola

Question

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

\[\dfrac{4}{3}\].
What is the equation of the new parabola?
\[y=\]

Answer 1

To begin with, the standard equation of the parabola is given by

\[ y = x^2. \]

Step 1: Reflecting across the \(x\)-axis

When we reflect a function across the \(x\)-axis, we negate the \(y\) values of the function. Therefore, the equation of the parabola after reflecting across the \(x\)-axis becomes:

\[ y = -x^2. \]

Step 2: Scaling Vertically

Next, we need to scale the parabola vertically by a factor of \(\frac{4}{3}\). Scaling vertically means multiplying the \(y\) value of the function by the scaling factor. Therefore, we multiply the equation we have after reflection by \(\frac{4}{3}\):

\[ y = \frac{4}{3}(-x^2). \]

This simplifies to:

\[ y = -\frac{4}{3} x^2. \]

Conclusion

Thus, the equation of the new parabola after reflection across the \(x\)-axis and a vertical scaling by a factor of \(\frac{4}{3}\) is:

\[ \boxed{-\frac{4}{3} x^2}. \]