Question

image shows f (x) = √x-2
and points (2,0) (3, 1) (6, 2)

Which table shows that y=−f(x) results in a reflection over the x-axis? (1 point)

Answer 1

To demonstrate how the function \( y = -f(x) \) represents a reflection of the function \( f(x) = \sqrt{x} - 2 \) over the x-axis, we can calculate the values of \( y \) for the points you provided and then create a table to show the relationship.

Given points from the function \( f(x) \):
1. For \( x = 2 \):
\[
f(2) = \sqrt{2} - 2 \approx 0, \quad y = -f(2) = -0 = 0
\]
2. For \( x = 3 \):
\[
f(3) = \sqrt{3} - 2 \approx 1 \quad y = -f(3) \approx -1
\]
3. For \( x = 6 \):
\[
f(6) = \sqrt{6} - 2 \approx 2 \quad y = -f(6) \approx -2
\]

Now, let's organize this information into a table:

| \( x \) | \( f(x) \) | \( -f(x) \) |
|---------|----------------|-----------------|
| 2 | \( 0 \) | \( 0 \) |
| 3 | \( 1 \) | \( -1 \) |
| 6 | \( 2 \) | \( -2 \) |

This table shows how reflecting the points over the x-axis changes the sign of the \( y \)-values of the original function \( f(x) \). The points \( (2, 0), (3, 1), (6, 2) \) reflect to \( (2, 0), (3, -1), (6, -2) \).