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 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 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 negative 1 2 negative 2 0 negative 3 2

x g(x)
1 2
2 0
3 2

Answer 1

To determine the correct table for \( g(x) = -f(-x) \), we need to first analyze the original function \( f(x) = 2(x + 22) \) and use some points from that function to find the corresponding points for \( g(x) \).

### Step 1: Find points for \( f(x) \)

Using the given function, let's find some sample inputs and their corresponding outputs.

For example:
- When \( x = -1 \):
\[
f(-1) = 2(-1 + 22) = 2 \times 21 = 42
\]

- When \( x = 0 \):
\[
f(0) = 2(0 + 22) = 2 \times 22 = 44
\]

- When \( x = 1 \):
\[
f(1) = 2(1 + 22) = 2 \times 23 = 46
\]

Now we have the following points for \( f(x) \):
- \( (-1, 42) \)
- \( (0, 44) \)
- \( (1, 46) \)

### Step 2: Determine \( g(x) = -f(-x) \)

Now let's calculate \( g(x) \) for the same x-values:
- For \( x = -1 \):
\[
g(-1) = -f(1) = -46
\]

- For \( x = 0 \):
\[
g(0) = -f(0) = -44
\]

- For \( x = 1 \):
\[
g(1) = -f(-1) = -42
\]

This gives us the transformed points:
- \( (-1, -46) \)
- \( (0, -44) \)
- \( (1, -42) \)

However, it seems there might be some discrepancies on using the points based on a direct function since the function to analyze should reflect an existing graph.

### Final Table Selection

Looking closely at the provided tables, we need purely to identify which corresponds to the correct transformation.

By checking each:
- The specific option that contains negative values corresponding to the reflection over the origin would be the one with pairs like \( (1, -2) \), \( (2, 0) \), \( (3, -2) \).

Thus, the correct response is:

**x** | **g(x)**
--- | ---
1 | -2
2 | 0
3 | -2

This clearly shows a reflection across the origin aligned with negative outputs.