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 function f(x)=2(x+22)
is shown in the graph. Which table shows that g(x)=−f(−x)
results in a reflection about the origin of the function graphed?

Hint: Make a table of the x
- and y
-coordinates based on the graph. Use these points to determine the coordinates of the function reflected about the origin.

(1 point)
Responses

x g(x)
1 2
2 0
3 2
x f left parenthesis x right parenthesis 1 2 2 0 3 2

x g(x)
−1
−2
−2
0
−3
−2
x f left parenthesis x right parenthesis negative 1 negative 2 negative 2 0 negative 3 negative 2

x g(x)
1 −2
2 0
3 −2
x f left parenthesis x right parenthesis 1 negative 2 2 0 3 negative 2

x g(x)
−1
2
−2
0
−3
2
x f left parenthesis x right parenthesis negative 1 2 negative 2 0 negative 3 2
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Answer 1

To determine which table represents the function \( g(x) = -f(-x) \) resulting in a reflection about the origin of the function graphed, we first must analyze how to derive \( g(x) \) based on the points from the parabola given by \( f(x) = 2(x+22) \).

In order to reflect a point about the origin, if we have a point \( (x, f(x)) \), the reflected point will be \( (-x, -f(x)) \).

Let's assume we have the following points from the graph of \( f(x) \):
- For \( x = 1 \), \( f(1) = 2 \), so the reflected point is \( (-1, -2) \).
- For \( x = 2 \), \( f(2) = 0 \), so the reflected point is \( (-2, 0) \).
- For \( x = 3 \), \( f(3) = 2 \), so the reflected point is \( (-3, -2) \).

Now, organizing these reflected points for \( g(x) = -f(-x) \):
- When \( x = 1 \), \( g(1) = -f(-1) \) leads to \( (-1, -2) \).
- When \( x = 2 \), \( g(2) = -f(-2) \) leads to \( (-2, 0) \).
- When \( x = 3 \), \( g(3) = -f(-3) \) leads to \( (-3, -2) \).

Now, we can write these points down in terms of \( g(x) \):
- For \( x = -1 \), \( g(-1) = 2 \) (from the reflected point)
- For \( x = -2 \), \( g(-2) = 0 \) (from the reflected point)
- For \( x = -3 \), \( g(-3) = 2 \) (from the reflected point)

Based on the provided choices, the corresponding table for the function \( g(x) = -f(-x) \) should have:
- \( g(1) = -2 \)
- \( g(2) = 0 \)
- \( g(3) = -2 \)

The correct table that reflects these points would be:

\[
\begin{array}{c|c}
x & g(x) \\
\hline
1 & -2 \\
2 & 0 \\
3 & -2 \\
\end{array}
\]

This matches the table shown in your provided options:

- \( x \) \( g(x) \)
- 1 -2
- 2 0
- 3 -2

Therefore, that would be the answer.