To find a sequence of transformations that takes the graph of $y = 3 + 2\sqrt{4 - x}$ to the graph of $y = -\sqrt{x} + 6$, we need to analyze how the two functions relate to each other and identify the changes in their form.

Step 1: Understand the basic form of both equations

1. Starting function: $y = 3 + 2\sqrt{4 - x}$ This function is a transformation of the basic square root function $y = \sqrt{x}$. Here's what happens to the basic square root graph:

   • The $\sqrt{4 - x}$ suggests a horizontal reflection and shift.
   • The factor $2$ vertically stretches the graph by a factor of 2.
   • The constant $+3$ shifts the graph upward by 3 units.

2. Target function: $y = -\sqrt{x} + 6$ This is also a transformation of $y = \sqrt{x}$:

   • The negative sign in front of the square root reflects the graph vertically.
   • The constant $+6$ shifts the graph upward by 6 units.

Step 2: Step-by-step transformation process

1. Start with $y = 3 + 2\sqrt{4 - x}$:

   This function involves a horizontal reflection and a shift, so we begin by analyzing those:

2. Change from $\sqrt{4 - x}$ to $\sqrt{x}$:

   The expression $\sqrt{4 - x}$ represents a horizontal reflection and shift:

   • Replace $4 - x$ with $x$ to get $\sqrt{x}$, which is a shift of the graph to the right by 4 units (since $4 - x$ reflects it over the vertical axis).

   This can be achieved by a transformation:

   • Translation right by 4 units: Shift the graph of $y = 3 + 2\sqrt{4 - x}$ right by 4 units to get the graph of $y = 3 + 2\sqrt{x}$.

3. Vertical transformation (stretching and flipping):

   Now, focus on the coefficient of $2$ in $2\sqrt{x}$. The factor of 2 vertically stretches the graph by a factor of 2:

   • Vertical stretch by a factor of 2: You will stretch the graph of $y = 3 + \sqrt{x}$ by a factor of 2 to get $y = 3 + 2\sqrt{x}$.

4. Reflection and shift to the target form:

   Next, we need to adjust for the negative sign and vertical shift in the target equation:

   • Reflection about the x-axis: The negative sign in $-\sqrt{x}$ suggests a vertical reflection. Reflect the graph of $y = 3 + 2\sqrt{x}$ to get $y = 3 - 2\sqrt{x}$.
   • Shift up by 6 units: Finally, add 6 to shift the graph upward by 6 units, resulting in $y = -2\sqrt{x} + 6$.

5. Adjust the vertical stretch factor:

   • The target function $y = -\sqrt{x} + 6$ has a coefficient of 1 for $\sqrt{x}$, so you need to compress vertically by a factor of 2 (to undo the previous vertical stretch).

Final sequence of transformations:

1. Translate the graph of $y = 3 + 2\sqrt{4 - x}$ right by 4 units.
2. Vertically stretch the graph by a factor of 2.
3. Reflect the graph about the x-axis.
4. Shift the graph up by 6 units.
5. Compress the graph vertically by a factor of 2.

Would you like me to explain any step in more detail, or would you like help with another concept? Here are some related questions for further practice:

1. How do horizontal and vertical translations affect the graph of a function?
2. What is the effect of stretching or compressing a graph vertically and horizontally?
3. How does reflecting a graph affect its equation?
4. What happens when we combine multiple transformations on the same graph?
5. How do you identify the key transformations from two given functions?

Tip: When dealing with square root transformations, it helps to first focus on identifying the shifts (horizontal and vertical) and then handle the stretches, reflections, or compressions step by step.