The function f(x)= 2(x+2^2) is shown on graph with points (-2,0), (-1,2) and (-3,2) Which table shows that g(x). =-f(-x) results in a reflection about the origin of the function graphed

Question

x g(x)
1 2
2 0
3 2

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

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

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

Answer 1

To find what the function \( g(x) = -f(-x) \) will look like, we first need to analyze the original function \( f(x) = 2(x + 2^2) \).

1. Simplifying the function: \[ f(x) = 2(x + 4) = 2x + 8 \]

2. We will compute \( g(x) = -f(-x) \): \[ f(-x) = 2(-x + 4) = -2x + 8 \] Thus, \[ g(x) = -(-2x + 8) = 2x - 8 \]

Now, we can calculate \( g(x) \) for the values \( x = 1, 2, \) and \( 3 \):

• For \( x = 1 \): \[ g(1) = 2(1) - 8 = 2 - 8 = -6 \]
• For \( x = 2 \): \[ g(2) = 2(2) - 8 = 4 - 8 = -4 \]
• For \( x = 3 \): \[ g(3) = 2(3) - 8 = 6 - 8 = -2 \]

Thus, we find: \[ g(1) = -6, \quad g(2) = -4, \quad g(3) = -2 \]

Now let's look at the tables provided to see which one reflects this behavior. We can see that:

1. First table:

   • \( g(1) = 2 \)
   • \( g(2) = 0 \)
   • \( g(3) = 2 \)

2. Second table:

   • \( g(1) = -2 \)
   • \( g(2) = 0 \)
   • \( g(3) = -2 \)

3. Third table:

   • \( g(-1) = -2 \)
   • \( g(-2) = 0 \)
   • \( g(-3) = -2 \)

4. Fourth table:

   • \( g(-1) = 2 \)
   • \( g(-2) = 0 \)
   • \( g(-3) = 2 \)

None of the provided tables match the calculated values \( g(1) = -6 \), \( g(2) = -4 \), and \( g(3) = -2 \).

However, if we were to summarize the correct form of how \( g(x) = -f(-x) \) transforms the points, we could infer that it would provide a reflection about the origin when we consider the transformations in the context of the original points of the graph of \( f(x) \).

Since none of the options provided correctly reflect the calculations.

Please double check the setup or provide clarification on the specific points or transformations you would like to focus on.

Answer 2

x is 1,2,3 g(x) is 2,0,2 first table
x is 1,2,3. g(x) is -2 0, -2 in second table
x is -1,-2 -3, g(x) is-2,0,-2 third table
x is -1,-2,-3. g(x) is 2,0,2 fourth table

Answer 3

Thanks for your clarification regarding the tables. Let's break down the function \( g(x) = -f(-x) \) once again, focusing specifically on the correct values for \( g(x) \) as per \( f(x) \) and how they relate to reflections and transformations.

We earlier simplified \( f(x) = 2(x + 4) = 2x + 8 \).

Performing the Transformation to Find \( g(x) \):

We compute:

1. Find \( f(-x) \): \[ f(-x) = 2(-x + 4) = -2x + 8 \]

2. Then, find \( g(x) = -f(-x) \): \[ g(x) = -(-2x + 8) = 2x - 8 \]

Calculate \( g(x) \) for \( x = 1, 2, 3 \):

• For \( x = 1 \): \[ g(1) = 2(1) - 8 = 2 - 8 = -6 \]

• For \( x = 2 \): \[ g(2) = 2(2) - 8 = 4 - 8 = -4 \]

• For \( x = 3 \): \[ g(3) = 2(3) - 8 = 6 - 8 = -2 \]

Summary of \( g(x) \) values:

• \( g(1) = -6 \)
• \( g(2) = -4 \)
• \( g(3) = -2 \)

Analyzing the Tables:

Now let's analyze the provided tables to determine which one shows that \( g(x) \) behaves as a reflection about the origin.

1. First table:

   • \( g(1) = 2 \)
   • \( g(2) = 0 \)
   • \( g(3) = 2 \)

2. Second table:

   • \( g(1) = -2 \)
   • \( g(2) = 0 \)
   • \( g(3) = -2 \)

3. Third table:

   • \( g(-1) = -2 \)
   • \( g(-2) = 0 \)
   • \( g(-3) = -2 \)

4. Fourth table:

   • \( g(-1) = 2 \)
   • \( g(-2) = 0 \)
   • \( g(-3) = 2 \)

Reflection About the Origin

In order for \( g(x) = -f(-x) \) to represent a reflection about the origin, the outputs for positive \( x \) must match the outputs of negative \( x \), taking into account the sign change.

• Observing the Second table:
  • \( g(1) = -2 \)
  • \( g(3) = -2 \)

This table has negative outputs for positive inputs, which indicates reflection but does not match our calculations exactly.

Conclusion

Based on all the calculations and analyses:

• The second table does show that \( g(x) \) produces negative values for \( x \), which reflects characteristics of \( -f(-x) \).
• However, keep in mind that the provided values from calculations do not directly match any table as expected for \( x = 1, 2, 3 \).

From our reflection analysis, the table showing outputs that correspond to a transformation (though not exact) would be:

• Second table: \( g(1) = -2, g(2) = 0, g(3) = -2 \)

If you need any further clarification or additional tasks, feel free to ask!