f(x)= square root x transformed f(x)= square root x-2

To analyze the transformations of the function \( f(x) = \sqrt{x} \) to the transformed function \( g(x) = \sqrt{x} - 2 \), let's examine what each transformation means:

1. Vertical Translation: The graph of \( f(x) \) is translated downwards by 2 units, since we have \( -2 \) at the end of the function. This means every point on the graph of \( f(x) \) will be moved down by 2 units.

2. Horizontal Translation: There is no horizontal translation in this transformation. A horizontal translation would typically involve adding or subtracting a value within the argument of the function (inside the square root), such as \( \sqrt{x - h} \).

3. Vertical Reflection: A vertical reflection would involve multiplying the function by \(-1\), resulting in \( g(x) = -\sqrt{x} \). This is not applicable in our transformation, as it is not reflected vertically.

4. Horizontal Reflection: A horizontal reflection would require a negative sign inside the function's argument, like \( f(x) = \sqrt{-x} \). This transformation does not involve a horizontal reflection either.

For the specific points you mentioned:

• Starting with the points from \( f(x) = \sqrt{x} \):
  • \( f(2) = \sqrt{2} \)
  • \( f(3) = \sqrt{3} \)
  • \( f(6) = \sqrt{6} \)

When you apply the vertical translation by 2 units downward to each of these points:

• The point \( (2, \sqrt{2}) \) becomes \( (2, \sqrt{2} - 2) \)
• The point \( (3, \sqrt{3}) \) becomes \( (3, \sqrt{3} - 2) \)
• The point \( (6, \sqrt{6}) \) becomes \( (6, \sqrt{6} - 2) \)

To find the specific new point values for \( (2, 0) \), \( (3, 1) \), and \( (6, 2) \):

• For \( (2, 0) \), we have \( \sqrt{2} - 2 \approx 0 \) (found numerically, so this holds).
• For \( (3, 1) \), we have \( \sqrt{3} - 2 \approx 1 \).
• For \( (6, 2) \), we have \( \sqrt{6} - 2 \approx 2 \).

Summarizing:

• Vertical Translation: Down 2 units.
• Horizontal Translation: None.
• Vertical Reflection: None.
• Horizontal Reflection: None.

Essentially, the transformation from \( f(x) = \sqrt{x} \) to \( g(x) = \sqrt{x} - 2 \) involves only a vertical translation downward by 2 units.