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 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

Answer 1

To determine which table shows that \( g(x) = -f(-x) \) results in a reflection about the origin of the function graphed, we first need to analyze how the function \( f(x) = 2(x + 22) \) behaves.

From the function \( g(x) = -f(-x) \), let's find the values for \( g(x) \):

1. Substituting \( -x \) into \( f(x) \): \[ f(-x) = 2(-x + 22) = 2(-x) + 44 = -2x + 44 \]

2. Now, to find \( g(x) \): \[ g(x) = -f(-x) = -(-2x + 44) = 2x - 44 \]

Given the nature of reflections about the origin, if a point \((a, b)\) lies on \( f(x) \), the corresponding point \((-a, -b)\) should lie on \( g(x) = -f(-x) \).

Now let's analyze the given tables to check for these pairs \((a, b)\) and \((-a, -b)\):

1. First table:

   • Points: \( (1, 2), (2, 0), (3, 2) \)
   • Reflection about the origin: \( (-1, -2), (-2, 0), (-3, -2) \)

2. Second table:

   • Points: \( (-1, -2), (-2, 0), (-3, -2) \)
   • We can't determine the reflection for this table without the positive \( x \) values.

3. Third table:

   • Points: \( (1, -2), (2, 0), (3, -2) \)
   • Reflection about the origin: \( (-1, 2), (-2, 0), (-3, 2) \)

4. Fourth table:

   • Points: \( (-1, 2), (-2, 0), (-3, 2) \)
   • Similar to the second table, we can't determine the reflection for this table without the positive \( x \) values.

From the analysis:

• First table represents points and doesn't match with origin reflection.
• Second table could be considered reflected but lacks the corresponding \(x\) values.
• Third table does not represent a reflection about the origin because points are not negated correctly.
• Fourth table, again, does not display the required reflection format.

However, we need to find where pairs meet the criteria for reflection \( g(x) = -f(-x) \). Upon reviewing values:

The accurate matched table showing the reflection of points clearly aligns with:

The Third Table:

• Correctly verifying: \( (1, -2) \) reflects to \( (-1, 2) \) etc aligning to the original function.

So, the table that correctly depicts reflection about the origin is the third one:

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

Thus, the third table corresponds to \( g(x) = -f(-x) \), showing the necessary reflection properties.