To transform $f(x) = x^2$ into $g(x) = -4x^2$, the following transformations took place:

1. Vertical Reflection: The negative sign ($-$) in front of the quadratic term reflects the graph of $f(x)$ across the x-axis. This flips the parabola so that it opens downwards instead of upwards.

2. Vertical Stretch: The coefficient 4 causes a vertical stretch of the graph by a factor of 4. This makes the parabola narrower compared to $f(x) = x^2$.

Summary of Transformations:

• Reflection across the x-axis.
• Vertical stretch by a factor of 4.

Would you like further clarification or a graphical comparison? Here are 5 related questions to deepen your understanding:

1. How would the transformations change if the function were $g(x) = 4x^2$?
2. What effect does a horizontal reflection have on $f(x) = x^2$?
3. How does adding a constant (e.g., $g(x) = -4x^2 + 5$) affect the graph?
4. What is the difference between a vertical stretch and a vertical compression?
5. How would transformations differ if $g(x) = -4(x - 2)^2 + 3$?

Tip: When analyzing transformations, start with the most dominant changes (like reflections or stretches) before considering translations.