We are given the function $f(x) = -(x+3)^2 - 2$ and are tasked with applying a series of transformations to it in order to find the function $g(x)$.

Step-by-step transformation process:

1. Translation 2 units to the right:

   • When translating a function horizontally, we adjust the $x$-term inside the function. A translation 2 units to the right means we subtract 2 from $x$.
   • This results in the transformation: $f(x-2) = -((x-2)+3)^2 - 2 = -(x+1)^2 - 2.$

2. Translation 2 units up:

   • To translate a function vertically, we add to or subtract from the function's output (the $y$-value). A translation 2 units up means we add 2 to the function.
   • Applying this to $-(x+1)^2 - 2$: $g(x) = -(x+1)^2 - 2 + 2 = -(x+1)^2.$

3. Reflection across the y-axis:

   • To reflect a function across the y-axis, we replace $x$ with $-x$ in the function.
   • Applying this reflection to $-(x+1)^2$: $g(x) = -(-x+1)^2 = -(1 - x)^2.$

Thus, the rule for $g(x)$ after applying all the transformations is: $g(x) = -(1 - x)^2.$

Would you like more details on any of these steps or have further questions?

Related Questions:

1. How do you graph the function $g(x) = -(1 - x)^2$?
2. What effect does each transformation have on the graph of a function?
3. How do reflections across the x-axis and y-axis differ?
4. How do you determine the vertex of a parabola after translation?
5. How can we check if two functions are reflections of each other?

Tip: When reflecting a function across the y-axis, only the $x$-variable changes sign, not the entire function.