Question

transform the graph of f(x)=x^2 to the graph of -f(x+2)-3

Answer 1

To transform the graph of the function \( f(x) = x^2 \) to the graph of \( -f(x+2)-3 \), we will follow the transformations step by step:

1. **Horizontal Translation**:
- \( f(x + 2) \) shifts the graph of \( f(x) \) two units to the left.
- So, starting with \( f(x) = x^2 \), we have:
\[
f(x + 2) = (x + 2)^2
\]

2. **Vertical Reflection**:
- The negative sign in front of \( f(x + 2) \) indicates a reflection over the x-axis.
- Therefore, we have:
\[
-f(x + 2) = - (x + 2)^2
\]

3. **Vertical Translation**:
- Finally, the term \(-3\) indicates a downward shift of 3 units.
- Thus, we have:
\[
-f(x + 2) - 3 = - (x + 2)^2 - 3
\]

Combining all these transformations, the final transformed function is:
\[
g(x) = - (x + 2)^2 - 3
\]

### Summary of the Transformations:
- The original function \( f(x) = x^2 \) is first shifted 2 units to the left.
- It is then reflected over the x-axis.
- Lastly, it is shifted down by 3 units.

This process will result in a new parabola that opens downward, with its vertex moved to the new location. To find the vertex of \( g(x) \), we can determine that the original vertex \( (0, 0) \) from the parabola \( f(x) = x^2 \) shifts to \( (-2, -3) \) after applying all transformations.