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 −2x 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 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 negative 1 2 negative 2 0 negative 3 2
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Answer 1

To determine the correct table reflecting \( g(x) = -f(-x) \), we first need to understand the relationship between the original function \( f(x) = 2(x + 22) \) and its transformed version \( g(x) \).

1. Reflection across the y-axis (\( -x \)): This means we will evaluate \( f(-x) \).
2. Reflection across the x-axis: After that, to find \( g(x) \), we will negate the result of \( f(-x) \).

To see how these transformations work, let's start with some example values to create the table based on reflection about the origin.

Assuming we have a table for \( f(x) \):

| x | f(x) | |------|--------| | 1 | 2 | | 2 | 0 | | 3 | -2 |

Now let's compute \( g(x) = -f(-x) \):

• For \( x = 1 \):

  • \( f(-1) = f(1) = 2 \) (from table)
  • So, \( g(1) = -f(-1) = -2 \)

• For \( x = 2 \):

  • \( f(-2) = f(2) = 0 \) (from table)
  • So, \( g(2) = -f(-2) = 0 \)

• For \( x = 3 \):

  • \( f(-3) = f(3) = -2 \) (from table)
  • So, \( g(3) = -f(-3) = 2 \)

This gives us the following table for \( g(x) \):

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

Now, when we compare this with the options given, the answer would be:

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

This matches the table:

x	g(x)
1	−2
2	0
3	2 

So, the correct response is:

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