To understand how the function [tex]\( g(x) = \sqrt{x} - 4 \)[/tex] is derived from the function [tex]\( f(x) = \sqrt{x} \)[/tex], we need to examine the adjustment made to the function.
1. Original Function Analysis:
The original function is [tex]\( f(x) = \sqrt{x} \)[/tex]. This function produces a graph that starts at the origin (0,0) and rises gradually to form a curve that heads upward to the right.
2. Form of the Transformed Function:
The transformed function is [tex]\( g(x) = \sqrt{x} - 4 \)[/tex]. The transformation involves subtracting 4 from the entire function [tex]\( \sqrt{x} \)[/tex].
3. Effect of Subtracting a Constant:
When a constant is subtracted from the function, it affects the y-values of the function directly:
- [tex]\( g(x) = \sqrt{x} - 4 \)[/tex] means that for every [tex]\( x \)[/tex], the corresponding [tex]\( y \)[/tex]-value in [tex]\( f(x) \)[/tex] is reduced by 4 units.
4. Graphical Interpretation:
- Subtracting 4 from the function [tex]\( f(x) \)[/tex] causes the entire graph to shift downward by 4 units because every point on the curve [tex]\( f(x) = \sqrt{x} \)[/tex] is moved down 4 units.
So, the operation of subtracting 4 from the function [tex]\( f(x) \)[/tex] results in a vertical shift downward by 4 units.
Therefore, the correct option is:
C) Translates the graph 4 units downward
