The given function is $g(x) = -\sqrt{x - 3} + 2$. To describe the transformations from the parent graph $f(x) = \sqrt{x}$, we break down the changes step by step:

1. Horizontal Translation (Right Shift):

   • The term inside the square root, $(x - 3)$, indicates a horizontal shift.
   • Since it's $x - 3$, the graph shifts 3 units to the right.

2. Reflection (Vertical Flip):

   • The negative sign in front of the square root, $-\sqrt{x - 3}$, reflects the graph across the x-axis. This means the graph will now point downward instead of upward.

3. Vertical Translation (Up Shift):

   • The $+2$ outside the square root shifts the graph 2 units up.

Summary of Transformations:

• The parent graph $f(x) = \sqrt{x}$ is first shifted 3 units to the right.
• It is then reflected across the x-axis (flipped upside down).
• Finally, it is shifted 2 units up.

Would you like more details on any of these transformations?

5 Related Questions:

1. How would the graph change if the function was $g(x) = \sqrt{x + 3}$?
2. What does a vertical stretch or compression look like in a square root function?
3. How can you graph $g(x) = -\sqrt{x - 3} + 2$ by hand?
4. What effect does a horizontal reflection (over the y-axis) have on the square root function?
5. How do horizontal translations differ from vertical translations in general?

Tip:

When analyzing transformations, always start with the operations inside the function (like shifts and reflections), then move to outside operations (like stretching and translations).