https://file.aiquickdraw.com/mathbot/file/1727225991182ucnyuc0v.jpg

The problem asks for the expression of the function \( g(x) \) when the graph of the function \( f(x) = x^2 \) is translated a whole number of units horizontally and vertically.

### Analysis of the translation:
1. **Horizontal translation**: From the graph, we can observe that the graph of \( g(x) \) has been shifted 2 units to the left. This horizontal shift is represented by \( x \to x + 2 \).
   
2. **Vertical translation**: The vertex of the parabola is now at \( ( -2, 0 ) \), compared to the vertex of \( f(x) = x^2 \) which is at \( (0, 0) \). It appears that the parabola has been reflected vertically (flipped) but not shifted vertically.

3. **Reflection**: The graph is reflected across the x-axis. This is represented by the negative sign in front of the expression.

### Expression for \( g(x) \):
Given these transformations (shift left by 2 and a reflection across the x-axis), the expression for \( g(x) \) is:

\[
g(x) = -(x + 2)^2
\]

This matches the given answer in the image.

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

Here are 5 related questions:
1. What effect does reflecting a parabola across the x-axis have on its equation?
2. How do you determine horizontal and vertical shifts from a graph?
3. How would the equation change if the graph was shifted 3 units up instead?
4. What would the equation of the function be if it was shifted 2 units right instead of left?
5. How do you find the vertex of a parabola from its equation in vertex form?

**Tip**: Always pay attention to the direction of shifts and reflections when transforming a graph!

The graph of f is translated a whole number of units horizontally and vertically to obtain the graph of g. The function f is defined by f(x) = x^2. Write down the expression for g(x).

Learn how to find the expression for the transformed quadratic function g(x) from the base function f(x) = x^2 using translations and reflections. This problem explores key concepts in function transformations, suitable for students in Grades 9-12.

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