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).