To understand the transformations from the function \( f(x) = \sqrt{x} \) to \( y = f(x + 5) - 4 \), let's break down the changes step by step.
1. Transformation of \( x \) to \( x + 5 \):
- The original function is \( f(x) = \sqrt{x} \).
- When we replace \( x \) with \( x + 5 \), the function becomes \( f(x + 5) = \sqrt{x + 5} \).
- This replacement means that the graph of the function is shifted horizontally. Specifically, since we are adding 5 inside the function argument, it shifts the graph to the left by 5 units.
2. Transformation of \( \sqrt{x + 5} \) to \( \sqrt{x + 5} - 4 \):
- Now consider the function \( \sqrt{x + 5} \), and then we subtract 4 from it: \( y = \sqrt{x + 5} - 4 \).
- Subtracting 4 from the entire function causes a vertical shift. Specifically, subtracting 4 from the function means we shift the graph downward by 4 units.
So, summarizing the transformations:
- There is a horizontal shift left by 5 units due to the substitution of \( x \) with \( x + 5 \).
- There is a vertical shift down by 4 units due to the subtraction of 4 from the function value.
Thus, the two transformations are:
1. Horizontal shift left by 5 units.
2. Vertical shift down by 4 units.
