The function given is $y = \sqrt{x - 4}$. This is a square root function that is shifted horizontally.

Steps to analyze the graph:

1. Basic Form: The base square root function is $y = \sqrt{x}$. The graph starts at the origin $(0,0)$ and increases gradually, only defined for non-negative $x$.

2. Shift of the Graph:

   • The expression inside the square root, $x - 4$, represents a horizontal shift. Specifically, it shifts the graph 4 units to the right. So instead of starting at $x = 0$, the graph now starts at $x = 4$.
   • There is no vertical shift, as there is no term added or subtracted outside of the square root function.

3. Domain: The square root function is only defined for $x \geq 4$ (since the expression inside the square root must be non-negative).

4. Range: Since the square root outputs only non-negative values, the range of the function is $y \geq 0$.

Identifying the correct graph:

The graph should start at the point $(4, 0)$ and increase gradually to the right.

• In the image, Graph B shows the point starting at $(4, 0)$ and then increases to the right, which matches the expected graph of $y = \sqrt{x - 4}$.

Answer:

The correct graph is B.

Would you like a more detailed breakdown of the steps or more examples?

Here are five related questions for you to consider:

1. How does the graph of $y = \sqrt{x + 4}$ differ from $y = \sqrt{x - 4}$?
2. What would happen to the graph if there was a vertical shift, such as $y = \sqrt{x - 4} + 2$?
3. What is the domain and range of the function $y = -\sqrt{x - 4}$?
4. How do you find the inverse of the function $y = \sqrt{x - 4}$?
5. How would the graph change if the function was $y = \sqrt{2x - 8}$?

Tip:

When graphing square root functions, always pay attention to shifts inside the square root (horizontal) and outside the square root (vertical).