Let the graph of g be a translation 2 units up and 2 units right, followed by a reflection in the y-axis of the graph of f(x)= -(x+3)^2 - 2. Write a rule for g.

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.

Let the graph of g be a translation 2 units up and 2 units right, followed by a reflection in the y-axis of the graph of f(x) = -(x+3)^2 - 2. Write a rule for g.

Learn how to transform the quadratic function f(x) = -(x+3)^2 - 2 by translating it 2 units right, 2 units up, and reflecting it across the y-axis. This detailed solution walks through each step and provides the rule for the transformed function g(x). Suitable for students in Grades 9-12.

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