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The problem is asking to graph the function \( h(x) = (x + 5)^2 + 4 \) using transformations of \( f(x) = x^2 \).

### Analyzing the function \( h(x) = (x + 5)^2 + 4 \):

1. The base function is \( f(x) = x^2 \), which is a standard parabola with its vertex at the origin (0,0).
2. The transformation \( (x + 5)^2 \) implies a **horizontal shift** to the **left** by 5 units.
3. The transformation \( +4 \) at the end represents a **vertical shift** **up** by 4 units.

Therefore, the function \( h(x) = (x + 5)^2 + 4 \) is the graph of \( f(x) = x^2 \), shifted 5 units to the left and 4 units upward. This matches **option A**: "The graph of \( f(x) = x^2 \) should be horizontally shifted to the left by 5 units and vertically shifted up by 4 units."

Would you like more details on graph transformations, or do you have any questions?

### Related Questions:
1. What is a horizontal shift in a graph?
2. How do vertical shifts affect the graph of a function?
3. Can transformations apply to other types of functions, like cubic or absolute value functions?
4. What is the vertex of a parabola, and how do shifts affect it?
5. How would you describe the transformation of \( f(x) = x^2 \) into \( h(x) = (x - 3)^2 - 2 \)?

**Tip:** Horizontal shifts are always in the opposite direction of the sign inside the function. For example, \( (x + 5) \) means a shift to the left, not the right!

Use transformations of f(x) = x^2 to graph the following function. h(x) = (x + 5)^2 + 4.

Learn how to use transformations to graph the quadratic function h(x) = (x + 5)^2 + 4. This problem involves shifting the basic parabola horizontally and vertically, important for understanding graph transformations in quadratic functions. Suitable for students in Grades 9-10.

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