Use the image to answer the question.

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 negative 1 2 negative 2 0 negative 3 2

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

Answer 1

To determine which table correctly represents the function \( g(x) = -f(-x) \) as a reflection about the origin of the function graphed, we first need to evaluate the points from the function \( f(x) \) given in the problem. The points from \( f(x) \) can be derived from the transformation applied to the x-coordinates and y-coordinates.

From the equation \( g(x) = -f(-x) \), we can derive the coordinates:

1. Take each x-coordinate from \( f(x) \).
2. Replace \( x \) with \( -x \).
3. Find \( f(-x) \).
4. Apply the negative to find \( g(x) \).

Given the values provided in the possible tables, we can evaluate:

Assuming the provided function points from the graph are:

• \( f(-1) \): let's say this equals \( 2 \)
• \( f(2) \): let's say this equals \( 0 \)
• \( f(-2) \): let's say this equals \( -2 \)
• \( f(0) \): let's say this equals \( 2 \)
• \( f(-3) \): let's say this equals \( 2 \)

Now, applying the transformation \( g(x) = -f(-x) \):

1. For \( x = -1 \):

   • \( g(-1) = -f(1) \) (if \( f(1) = 2 \)), then \( g(-1) = -2 \)

2. For \( x = 2 \):

   • \( g(2) = -f(-2) \) (if \( f(-2) = 0 \)), then \( g(2) = 0 \)

3. For \( x = -2 \):

   • \( g(-2) = -f(2) \) (if \( f(2) = -2 \)), then \( g(-2) = 2 \)

4. For \( x = 0 \):

   • \( g(0) = -f(0) \) (if \( f(0) = 2 \)), then \( g(0) = -2 \)

5. For \( x = 3 \):

   • \( g(3) = -f(-3) \) (if \( f(-3) = 2 \)), then \( g(3) = -2 \)

So reflecting about the origin, we'd have the points as follows:

• \( g(-1) = -2 \)
• \( g(2) = 0 \)
• \( g(-2) = 2 \)
• \( g(0) = -2 \)
• \( g(-3) = -2 \)

Based on the tables given, it looks like:

The correct answer appears to be:

x     g(x)
−1   −2  
−2   0  
−3   −2  

So the table that demonstrates \( g(x) = -f(-x) \) reflects the points from the function graphed in the origin (which is reflective through both axes) is the fourth option.