The problem asks to determine the equation of a transformed function based on the parent function $y = \sqrt{x}$, where the graph has undergone transformations, and one particular point, $(-2, 5)$, is indicated.

To solve this, we need to break down the transformations:

1. Horizontal Shift (left or right): The equation $y = \sqrt{x - h}$ represents a horizontal shift by $h$ units. If $h > 0$, the graph shifts right; if $h < 0$, the graph shifts left.

2. Vertical Shift (up or down): The equation $y = \sqrt{x} + k$ shifts the graph vertically by $k$ units. If $k > 0$, the graph moves up; if $k < 0$, the graph moves down.

3. The given point $(-2, 5)$ suggests a shift from the origin. To determine the equation, let's compare this point with the typical behavior of the square root function, which usually starts at the origin (0, 0).

Given the available choices:

1. $f(x) = \sqrt{x + 1} - 6$
2. $f(x) = \sqrt{x + 6} + 1$
3. $f(x) = \sqrt{x - 6} + 1$
4. $f(x) = \sqrt{x - 1} - 6$

Checking the first option:

• $f(x) = \sqrt{x + 1} - 6$ This implies a horizontal shift left by 1 unit and a vertical shift down by 6 units. When $x = -2$: $f(-2) = \sqrt{-2 + 1} - 6 = \sqrt{-1} - 6$ This result is invalid since the square root of a negative number is not real, so this option doesn't work.

Checking the second option:

• $f(x) = \sqrt{x + 6} + 1$ This implies a horizontal shift left by 6 units and a vertical shift up by 1 unit. When $x = -2$: $f(-2) = \sqrt{-2 + 6} + 1 = \sqrt{4} + 1 = 2 + 1 = 3$ This result doesn't match the point $(-2, 5)$, so this option is incorrect.

Checking the third option:

• $f(x) = \sqrt{x - 6} + 1$ This implies a horizontal shift right by 6 units and a vertical shift up by 1 unit. When $x = -2$: $f(-2) = \sqrt{-2 - 6} + 1 = \sqrt{-8} + 1$ This result is also invalid, so this option doesn't work.

Checking the fourth option:

• $f(x) = \sqrt{x - 1} - 6$ This implies a horizontal shift right by 1 unit and a vertical shift down by 6 units. When $x = -2$: $f(-2) = \sqrt{-2 - 1} - 6 = \sqrt{-3} - 6$ Again, this is invalid.

It seems the calculations do not correspond to the standard choices. Would you like to recheck or explore another approach?