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The problem provided asks to analyze two functions based on the base function \( f(x) = x^2 \). Let's break down each part:

### A) \( g(x) = (x + 3)^2 \)

This function is a transformation of the base function \( f(x) = x^2 \). Specifically, it represents a **horizontal shift** of the graph of \( f(x) \). The term \( (x + 3) \) shifts the graph **3 units to the left**.

- Horizontal transformations follow the rule \( f(x - h) \), where \( h \) determines the direction of the shift (left or right). Here, \( h = -3 \), so the shift is to the left.

### B) \( g(x) = x^2 + 3 \)

This function is a **vertical shift** of the base function \( f(x) = x^2 \). The addition of \( +3 \) outside the square moves the entire graph **3 units upward**.

- Vertical transformations occur when a constant is added outside of the base function, affecting the vertical position of the graph.

### Answers:
- A) \( g(x) = (x + 3)^2 \) is a **horizontal shift** to the left by 3 units.
- B) \( g(x) = x^2 + 3 \) is a **vertical shift** upward by 3 units.

Do you need further details on how transformations work? 

Here are 5 related questions to deepen understanding:
1. What would happen if \( g(x) = (x - 4)^2 \)?
2. How does adding a negative constant outside the function, like \( g(x) = x^2 - 2 \), affect the graph?
3. What is the effect of multiplying the function \( f(x) \) by a constant?
4. How does the transformation change if we have \( g(x) = (x + 3)^2 + 5 \)?
5. What is the difference between a horizontal stretch and a shift?

**Tip**: Horizontal transformations affect the input (inside the parentheses), while vertical transformations affect the output (outside the function).

Given the base function f(x) = x^2, what type of transformation are the functions g(x) = (x + 3)^2 and g(x) = x^2 + 3?

Analyze the transformations of the quadratic function f(x) = x^2 with horizontal and vertical shifts. Learn how g(x) = (x + 3)^2 results in a shift to the left by 3 units and how g(x) = x^2 + 3 shifts the graph up by 3 units. Suitable for high school algebra students.

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