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To solve this problem, we need to determine how the graph of the function \( f(x) = x^2 \) is translated to obtain the graph of \( h(x) \).

### Analyzing the Translation

The original function \( f(x) = x^2 \) is a parabola that opens upwards with its vertex at the origin (0, 0). 

Looking at the new graph (in blue), we see that the vertex of the parabola is moved to a new point. Let's identify the translation:

1. **Horizontal Shift**: The vertex of the blue parabola is located at \( x = 3 \).
   This indicates a horizontal shift to the right by 3 units.

2. **Vertical Shift**: The vertex of the blue parabola is located at \( y = -5 \).
   This indicates a vertical shift down by 5 units.

### Writing the Expression for \( h(x) \)

A translation of the graph horizontally by 3 units to the right and vertically by 5 units down means we modify the original function \( f(x) = x^2 \) as follows:

\[
h(x) = (x - 3)^2 - 5
\]

### Conclusion

The expression for the function \( h(x) \) after translating \( f(x) = x^2 \) is:

\[
\boxed{h(x) = (x - 3)^2 - 5}
\]

Would you like a detailed explanation of each step? 

### Related Questions:

1. How do horizontal and vertical shifts affect the equation of a quadratic function?
2. What are the general steps to find the vertex of a parabola given its equation?
3. How do we determine if a parabola opens upwards or downwards?
4. What is the impact of translating a function by negative values?
5. Can we translate a function diagonally? How would that look mathematically?

#### Tip:
To identify translations quickly, always look for changes in the vertex's position compared to the original function.

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

Explore how horizontal and vertical translations affect the graph of a quadratic function. This guide provides a detailed solution on how to express the function h(x) when f(x) = x^2 is shifted horizontally and vertically, using the vertex form of a quadratic equation.

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