Question

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 2 2 0 3 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 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 how the function \( g(x) = -f(-x) \) reflects the original function \( f(x) \) about the origin, we need to analyze how the function transforms.

1. Start with the original function:

   The function given is \( f(x) = 2(x + 2)^2 \).

2. Find points on the graph \( f(x) \):

   From the description, let's sample a few points:

   • If \( x = -2 \): \[ f(-2) = 2(-2 + 2)^2 = 2(0)^2 = 0 \]
   • If \( x = -1 \): \[ f(-1) = 2(-1 + 2)^2 = 2(1)^2 = 2 \]
   • If \( x = 0 \): \[ f(0) = 2(0 + 2)^2 = 2(2)^2 = 8 \]
   • If \( x = 1 \): \[ f(1) = 2(1 + 2)^2 = 2(3)^2 = 18 \]
   • If \( x = 2 \): \[ f(2) = 2(2 + 2)^2 = 2(4)^2 = 32 \]

   Thus, our table of points from \( f(x) \) looks like:

   | x | \( f(x) \) | |-----|------------| | -2 | 0 | | -1 | 2 | | 0 | 8 | | 1 | 18 | | 2 | 32 |

3. Calculate points for \( g(x) = -f(-x) \):

   To construct the \( g(x) \) values, we will plug in \( -x \) into \( f(x) \) and then take the negative of that result.

   • For \( x = -2 \): \[ g(-2) = -f(2) = -32 \]
   • For \( x = -1 \): \[ g(-1) = -f(1) = -18 \]
   • For \( x = 0 \): \[ g(0) = -f(0) = -8 \]
   • For \( x = 1 \): \[ g(1) = -f(-1) = -2 \]
   • For \( x = 2 \): \[ g(2) = -f(-2) = 0 \]

   Thus, the table for \( g(x) \) looks like:

   | x | \( g(x) \) | |-----|------------| | -2 | -32 | | -1 | -18 | | 0 | -8 | | 1 | -2 | | 2 | 0 |

4. Check response options:

To reflect about the origin, we need the coordinates in the form \( (x, y) \to (-x, -y) \):

From the computations above for \( g(x) \), if we look at our original points and their reflections:

• From \( (2, 32) \) to \( (-2, -32) \)
• From \( (1, 18) \) to \( (-1, -18) \)
• From \( (0, 8) \) to \( (0, -8) \)
• From \( (-1, 2) \) to \( (1, -2) \)
• From \( (-2, 0) \) to \( (2, 0) \)

The correct response for the transformation reflected about the origin needs to result in:

| x | \( g(x) \) | |-----|------------| | -1 | -2 | | -2 | 0 | | 0 | -8 | | 1 | -18 | | 2 | -32 |

Therefore, the correct answer that matches the reflection properties is:

• \( g(x) \) values: \( -1, -2 \), \( -2, 0 \), \( 0, -8 \), \( 1, -18 \), \( 2, -32 \).

This corresponds to:

• \( g(x) \) \( 1, -2 \) \( 2, -0 \) \( 3, -2 \)...

Please follow the response format and match your table accordingly.