To describe how to transform the graph of $f(x) = \sqrt{x - 8}$ into the graph of $g(x) = \sqrt{x + 4}$, we need to understand how the transformation affects the input $x$.

Step-by-Step Analysis:

1. Starting with the basic function: The function $f(x) = \sqrt{x - 8}$ is a square root function shifted horizontally.

2. Horizontal Shifts:

   • For $f(x) = \sqrt{x - 8}$, the graph of $\sqrt{x}$ is shifted 8 units to the right because the expression inside the square root is $x - 8$.
   • For $g(x) = \sqrt{x + 4}$, the graph of $\sqrt{x}$ is shifted 4 units to the left because of the $x + 4$ inside the square root.

3. Transformation Description: To go from the graph of $f(x) = \sqrt{x - 8}$ to $g(x) = \sqrt{x + 4}$, you must shift the graph 12 units to the left. This is because you are moving from $x - 8$ to $x + 4$, a total shift of $8 + 4 = 12$ units left.

Summary:

• The transformation from $f(x) = \sqrt{x - 8}$ to $g(x) = \sqrt{x + 4}$ is a horizontal shift of 12 units to the left.

Do you need any further details or clarifications?

Here are 5 related questions you might explore:

1. How would the transformation change if we had $g(x) = \sqrt{x - 4}$?
2. What happens if we add a vertical shift, such as $g(x) = \sqrt{x + 4} + 2$?
3. How does the graph of $f(x) = \sqrt{x}$ compare to $g(x) = \sqrt{x + 4}$?
4. How would the transformation change if we considered vertical stretches or compressions?
5. How would you describe the domain of both $f(x)$ and $g(x)$?

Tip: Always remember that changes inside the function (like $x + 4$) affect the graph horizontally, while changes outside (like $\sqrt{x} + 2$) affect it vertically.