To reflect the function \( f(x) = (x+1)^2 \) about the origin, we need to determine the reflected function. The reflection of a function \( f(x) \) about the origin is given by \( -f(-x) \).
First, let's find \( -f(-x) \):
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Finding \( f(-x) \): \[ f(-x) = (-x + 1)^2 = (1 - x)^2 = 1 - 2x + x^2. \]
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Now, find \( -f(-x) \): \[ -f(-x) = - (1 - 2x + x^2) = -1 + 2x - x^2. \]
So the reflected function is: \[ -f(-x) = 2x - x^2 - 1. \]
Now, we need to find the reflected position of the point \( (-4, 9) \). To do that, we first calculate \( f(-4) \):
- Calculate \( f(-4) \): \[ f(-4) = (-4 + 1)^2 = (-3)^2 = 9. \] Since the y-value of point \((-4, 9)\) corresponds to this \( f(-4) \), it checks out.
Next, we find the coordinates of the reflected point, which is given by substituting \(-4\) into \( -f(-x) \):
- Calculate \( -f(-(-4)) = -f(4) \): \[ f(4) = (4 + 1)^2 = 5^2 = 25. \] Therefore, \[ -f(4) = -25. \]
So the reflected point of \( (-4, 9) \) in the function \( -f(-x) \) is \( (-4, -25) \).
Thus, the answer is: The point \( (-4, 9) \) reflects to \( (-4, -25) \) in \( -f(-x) \).