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To solve this problem, we need to transform the basic function \( f(x) = x^2 \) to match the function \( h(x) = (x + 4)^2 - 4 \). Let's break this down:

The function \( h(x) = (x + 4)^2 - 4 \) involves two transformations applied to the basic parabola \( f(x) = x^2 \):

1. **Horizontal Shift**: The term \( (x + 4) \) means that the graph is shifted 4 units to the left. This is because adding to \( x \) inside the parentheses results in a leftward shift.

2. **Vertical Shift**: The \( -4 \) outside the square indicates a shift 4 units downward, as subtracting a constant outside the function shifts the graph downward by that amount.

Now let's look at the answer choices to see which transformations are needed:

- **A. Shrink the graph vertically by a factor of 4**: This is incorrect; there is no vertical shrink.
- **B. Shrink the graph horizontally by a factor of 4**: This is incorrect; no horizontal shrinking is needed.
- **C. Stretch the graph horizontally by a factor of 4**: This is incorrect; no horizontal stretching is required.
- **D. Shift the graph 4 units down**: This is correct. The \( -4 \) outside the function shifts the graph down by 4 units.
- **E. Reflect the graph about the x-axis**: This is incorrect; there is no reflection.
- **F. Reflect the graph about the y-axis**: This is incorrect; there is no reflection across the y-axis.
- **G. Shift the graph 4 units to the left**: This is correct. The \( (x + 4) \) inside the parentheses shifts the graph 4 units to the left.
- **H. Stretch the graph vertically by a factor of 4**: This is incorrect; no vertical stretching occurs.
- **I. Shift the graph 4 units up**: This is incorrect; the graph shifts down, not up.
- **J. Shift the graph 4 units to the right**: This is incorrect; the shift is to the left.

### Correct transformations:
- **D. Shift the graph 4 units down.**
- **G. Shift the graph 4 units to the left.**

Would you like further clarification on any of these steps or transformations? 

Here are 5 related questions for practice:
1. How would the graph change if the function were \( h(x) = (x - 3)^2 + 2 \)?
2. What happens if the function includes a coefficient, like \( h(x) = 2(x + 4)^2 - 4 \)?
3. How does a vertical stretch differ from a vertical shrink?
4. What kind of transformation reflects the graph about the y-axis?
5. How does adding a constant inside the function \( f(x) = (x + k)^2 \) affect the graph?

**Tip**: Remember that shifts inside parentheses affect the graph horizontally (left or right), while changes outside the function affect the graph vertically (up or down).

Use transformations of f(x) = x^2 to graph the following function: h(x) = (x + 4)^2 - 4. Select all the transformations that are needed to graph the given function using f(x) = x^2.

Learn how to graph the function h(x) = (x + 4)^2 - 4 using transformations of the basic quadratic function f(x) = x^2. This problem covers key concepts such as horizontal and vertical shifts. Suitable for students in Grades 9-11.

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