Question

Reflect the function \(f(x) =(x +1)^2\) about the origin. What is the position of the point \((-4,9)\) in the reflected function?

Answer 1

To reflect the function \( f(x) = (x + 1)^2 \) about the origin, we will replace \( f(x) \) with \( -f(-x) \).

1. First, let's find \( f(-x) \):
\[
f(-x) = (-x + 1)^2 = (1 - x)^2
\]

2. Now we need to compute \( -f(-x) \):
\[
-f(-x) = -(1 - x)^2
\]
So the reflected function is:
\[
g(x) = -(1 - x)^2
\]

Next, we can simplify \( g(x) \):
\[
g(x) = -(1 - 2x + x^2) = -1 + 2x - x^2
\]
So the reflected function can be expressed as:
\[
g(x) = -x^2 + 2x - 1
\]

Now we want to find the position of the point \((-4, 9)\) in the reflected function. The original function \( f \) has the point \((-4, 9)\) since:
\[
f(-4) = (-4 + 1)^2 = (-3)^2 = 9.
\]

To find the corresponding point in the reflected function \( g \), we need to find \( g(4) \) (since reflecting about the origin means that we take the negative of both coordinates):
\[
g(4) = -4^2 + 2(4) - 1 = -16 + 8 - 1 = -9.
\]

Thus, the point \((-4, 9)\) in the original function corresponds to the point \((4, -9)\) in the reflected function.

Therefore, the position of the point \((-4, 9)\) in the reflected function is:
\[
(4, -9).
\]