Question

The parabola
\[y=x^2\] is shifted down by
\[3\] units and to the left by
\[2\] units.
What is the equation of the new parabola?
\[y=\]

Answer 1

To find the equation of the new parabola after shifting the original parabola \(y = x^2\) down by \(3\) units and to the left by \(2\) units, we need to adjust the equation accordingly.

1. **Shift down by 3 units**:
Shifting a function down by \(k\) units means subtracting \(k\) from the output (the \(y\) value). Therefore, shifting \(y = x^2\) down by \(3\) units gives:
\[
y = x^2 - 3
\]

2. **Shift to the left by 2 units**:
Shifting a function to the left by \(h\) units means replacing \(x\) with \(x + h\). In this case, we shift left by \(2\) units, so we replace \(x\) with \(x + 2\):
\[
y = (x + 2)^2 - 3
\]

Putting these two transformations together, we have the equation of the new parabola:
\[
y = (x + 2)^2 - 3
\]

Now, we can expand this equation if we wish:
\[
y = (x^2 + 4x + 4) - 3
\]
\[
y = x^2 + 4x + 1
\]

Thus, the equation of the new parabola can be expressed in either of the two following forms:
\[
y = (x + 2)^2 - 3 \quad \text{or} \quad y = x^2 + 4x + 1
\]

If we want a concise answer, we can write:
\[
\boxed{y = (x + 2)^2 - 3}
\]