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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+2)2
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 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

1 answer

To determine the coordinates for the function \( g(x) = -f(-x) \) based on the given function \( f(x) = 2(x+2)^2 \), we first need to identify some points from the graph of \( f(x) \) and then calculate the corresponding points for \( g(x) \).

Step 1: Identify Points from \( f(x) \)

Let's assume the points on the parabola \( f(x) \) are as follows based on the typical shape:

  1. If \( x = -2 \): \[ f(-2) = 2(-2+2)^2 = 2(0)^2 = 0 \] Point: \((-2, 0)\)

  2. If \( x = -1 \): \[ f(-1) = 2(-1+2)^2 = 2(1)^2 = 2 \] Point: \((-1, 2)\)

  3. If \( x = 0 \): \[ f(0) = 2(0+2)^2 = 2(2)^2 = 8 \] Point: \((0, 8)\)

  4. If \( x = 1 \): \[ f(1) = 2(1+2)^2 = 2(3)^2 = 18 \] Point: \((1, 18)\)

  5. If \( x = -3 \): \[ f(-3) = 2(-3+2)^2 = 2(-1)^2 = 2 \] Point: \((-3, 2)\)

Step 2: Calculate Points for \( g(x) \)

Now calculate \( g(x) \) using the formula \( g(x) = -f(-x) \):

  1. For \( x = 2 \): \[ g(2) = -f(-2) = -0 = 0 \] Point: \( (2, 0) \)

  2. For \( x = 1 \): \[ g(1) = -f(-1) = -2 \] Point: \( (1, -2) \)

  3. For \( x = 0 \): \[ g(0) = -f(0) = -8 \] Point: \( (0, -8) \)

  4. For \( x = -1 \): \[ g(-1) = -f(1) = -18 \] Point: \( (-1, -18) \)

  5. For \( x = -2 \): \[ g(-2) = -f(2) = -18 \] Point: \( (-2, -18) \)

Step 3: Match the Calculated Points to Options

From the options available:

  1. Choice 1: \[ \begin{array}{|c|c|} \hline x & g(x) \ \hline 1 & -2 \ 2 & 0 \ 3 & -2 \ \hline \end{array} \] This does match part of our calculations: \( g(1) = -2 \) and \( g(2) = 0 \).

  2. Choice 2: \[ \begin{array}{|c|c|} \hline x & g(x) \ \hline -1 & -2 \ 2 & 0 \ -2 & -3 \ 0 & 2 \ -3 & 2 \ \hline \end{array} \] This choice has inconsistencies.

  3. Choice 3: \[ \begin{array}{|c|c|} \hline x & g(x) \ \hline 1 & 2 \ 2 & 0 \ 3 & 2 \ \hline \end{array} \] This choice has inconsistencies.

  4. Choice 4: \[ \begin{array}{|c|c|} \hline x & g(x) \ \hline -1 & -2 \ -2 & 0 \ -3 & -2 \ \hline \end{array} \] This choice has inconsistencies.

The closest matching response that reflects the coordinates we calculated is Choice 1, but it’s crucial to ensure that other points are correct.

Thus, the best answer is: Choice 1: \[ \begin{array}{|c|c|} \hline x & g(x) \ \hline 1 & -2 \ 2 & 0 \ 3 & -2 \ \hline \end{array} \]