To transform the function $f(x) = x^2$ based on the given changes:

1. Flipped upside down: Reflecting the parabola across the x-axis changes the sign of the function, so it becomes $-x^2$.
2. Shifted to the left by 2 units: A horizontal shift to the left by 2 units means replacing $x$ with $(x + 2)$, giving $-(x + 2)^2$.
3. Shifted up by 2 units: A vertical shift upwards by 2 units involves adding 2 to the function, resulting in $-(x + 2)^2 + 2$.

Thus, the transformed function is:

$f(x) = -(x + 2)^2 + 2$

Would you like more details, or do you have any questions?

Here are 5 related questions to explore this concept further:

1. How would the graph change if the parabola were shifted to the right instead of the left?
2. What if the function were shifted down instead of up?
3. How does flipping the parabola affect its concavity?
4. What is the vertex of the transformed parabola $-(x + 2)^2 + 2$?
5. How would this transformation affect the x-intercepts of the function?

Tip: When transforming functions, always perform horizontal shifts before vertical shifts for clarity.