Question

The parabola
\[y=x^2\] is shifted up by
\[7\] units and to the left by
\[1\] unit.
What is the equation of the new parabola?
\[y=\]

Answer 1

To find the equation of the new parabola after a vertical shift and a horizontal shift, we can follow these steps:

1. Start with the original equation of the parabola:
\[
y = x^2
\]

2. Shift the parabola vertically upwards by 7 units. This means we add 7 to the output \(y\):
\[
y = x^2 + 7
\]

3. To shift the parabola to the left by 1 unit, we replace \(x\) with \(x + 1\) in the equation. This modification will account for the horizontal shift:
\[
y = (x + 1)^2 + 7
\]

4. Now, we can expand the equation:
\[
y = (x + 1)(x + 1) + 7
\]
\[
y = x^2 + 2x + 1 + 7
\]
\[
y = x^2 + 2x + 8
\]

Thus, the equation of the new parabola is:
\[
\boxed{y = x^2 + 2x + 8}
\]