Question

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)

Answer 1

To reflect a function \( f(x) \) about the origin, we need to apply the transformation \( g(x) = -f(-x) \). This means that we will take each point \( (x, f(x)) \) on the graph of \( f(x) \), negate the x-coordinate (replace \( x \) with \( -x \)), and then negate the resulting y-coordinate (multiply \( f(-x) \) by \( -1 \)).

First, let's find a few key points from the original function \( f(x) = 2(x + 22) \).

1. To find the points, we can choose some x-values:
- For \( x = -22 \): \( f(-22) = 2(-22 + 22) = 2(0) = 0 \) ➔ Point: \( (-22, 0) \)
- For \( x = -21 \): \( f(-21) = 2(-21 + 22) = 2(1) = 2 \) ➔ Point: \( (-21, 2) \)
- For \( x = -20 \): \( f(-20) = 2(-20 + 22) = 2(2) = 4 \) ➔ Point: \( (-20, 4) \)

Now, let's reflect these points about the origin to find corresponding points on the function \( g(x) = -f(-x) \):

- \( (x, f(x)) \) gives:
- \( (-22, 0) \) reflects to \( (22, 0) \)
- \( (-21, 2) \) reflects to \( (21, -2) \)
- \( (-20, 4) \) reflects to \( (20, -4) \)

Thus, the corresponding points on \( g(x) = -f(-x) \) will be:
- \( (22, 0) \)
- \( (21, -2) \)
- \( (20, -4) \)

Now we can summarize these into a table:
| x | f(x) | g(x) |
|-------|--------|--------|
| -22 | 0 | 0 |
| -21 | 2 | -2 |
| -20 | 4 | -4 |
| 20 | 0 | 0 |
| 21 | -2 | -2 |
| 22 | -4 | -4 |

This table isn't strictly necessary unless testing or comparing functions, but it demonstrates the transformation from \( f \) to \( g \) through point reflection about the origin.

Thus, the resulting table of coordinates for \( g(x) \) indicating a reflection about the origin from \( f(x) \) would ideally show:
| x | g(x) |
|-------|--------|
| 20 | -4 |
| 21 | -2 |
| 22 | 0 |

Select this option or create similar observations if the function needs visual reinforcement.