To reflect the function \( f(x) = (x+1)^2 \) about the origin, we need to take the negative of both the input and the output. The reflected function will be:
\[ -f(-x) = -((-x + 1)^2) = -(x - 1)^2. \]
Now let's evaluate the function at \( x = -4 \) to find the new position of the point \( (-4, 9) \).
First, we find \( -f(-(-4)) = -f(4) \).
Calculating \( f(4) \):
\[ f(4) = (4 + 1)^2 = 5^2 = 25. \]
Now calculate \( -f(4) \):
\[ -f(4) = -25. \]
Thus, the point \( (-4, 9) \) reflects to \( (-4, -25) \).
In conclusion, the point \( (-4, 9) \) reflects to \( (-4, -25) \) in the reflected function.