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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

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To determine how the function \( g(x) = -f(-x) \) reflects the original function \( f(x) = 2(x + 2)^2 \) about the origin, let's first understand the steps involved:

  1. Find \( f(-x) \): This means we substitute \(-x\) into the function \( f(x) \): \[ f(-x) = 2((-x) + 2)^2 = 2(-x + 2)^2 = 2(2 - x)^2 \]

  2. Find \( g(x) = -f(-x) \): \[ g(x) = -2(2 - x)^2 \]

Now, to reflect points about the origin, if \((x, y)\) is a point on the graph of function \( f \), then the corresponding point on the graph of function \( g \) will be \((-x, -y)\).

Next, we can create a table of coordinates based on the original function \( f(x) \) and then find the corresponding points for the reflected function \( g(x) \).

Example Points for \( f(x) \)

For instance, let's calculate \( f(x) \) at several values of \( x \):

  • If \( x = -4 \): \[ f(-4) = 2(-4 + 2)^2 = 2(-2)^2 = 2 \times 4 = 8 \quad \Rightarrow \quad (-4, 8) \]
  • If \( x = -2 \): \[ f(-2) = 2(-2 + 2)^2 = 2(0)^2 = 0 \quad \Rightarrow \quad (-2, 0) \]
  • If \( x = 0 \): \[ f(0) = 2(0 + 2)^2 = 2(2)^2 = 8 \quad \Rightarrow \quad (0, 8) \]
  • If \( x = 2 \): \[ f(2) = 2(2 + 2)^2 = 2(4)^2 = 32 \quad \Rightarrow \quad (2, 32) \]

Now reflecting these points about the origin:

  1. \((-4, 8) \rightarrow (4, -8)\)
  2. \((-2, 0) \rightarrow (2, 0)\)
  3. \((0, 8) \rightarrow (0, -8)\)
  4. \((2, 32) \rightarrow (-2, -32)\)

Table of Coordinates

Here is how the points look:

| \( x \) | \( f(x) \) | \( g(x) \) | |---------|------------|------------| | -4 | 8 | -8 | | -2 | 0 | 0 | | 0 | 8 | -8 | | 2 | 32 | -32 |

So you can use points from this table to validate that reflecting these points about the origin yields the points for the function \( g(x) = -f(-x) \).

The response you need may require selecting a corresponding table that matches the above points for \( g(x) \). Look for a table that shows coordinates where the values change signs compared to \( f(x) \).