Question

Use the image to answer the question.

A parabola is drawn on a coordinate plane. Both axes range from negative 5 to 5 in one-unit increments.

The graph shows the starting function y=2(x+2)2
. The graph of which of the following functions is a reflection of the starting function about the origin?

(1 point)
Responses

y=−2(x+2)2
y equals negative 2 left parenthesis x plus 2 right parenthesis squared

y=−2(−x+2)2
y equals negative 2 left parenthesis negative x plus 2 right parenthesis squared

y=2(−x+2)2
y equals 2 left parenthesis negative x plus 2 right parenthesis squared

y=2(x+2)2

Answer 1

To find the reflection of the function \( y = 2(x+2)^2 \) about the origin, we need to negate both \( x \) and \( y \). This means the transformation we apply is:

\[
y = -f(-x)
\]

For the function \( f(x) = 2(x+2)^2 \), we will first determine \( f(-x) \):

1. Substitute \(-x\) into the function:
\[
f(-x) = 2(-x + 2)^2
\]

2. Now, we need to negate this result for the reflection about the origin:
\[
y = -f(-x) = -2(-x + 2)^2
\]

Thus, the reflected function about the origin is:

\[
y = -2(-x + 2)^2
\]

From the options provided, the correct answer is:
\[
y = -2(-x + 2)^2
\]