To analyze the effect of the [tex]\(-4\)[/tex] on the function [tex]\(\sqrt{x}\)[/tex], consider the following step-by-step solution.
1. Understanding the Functions:
- The original function is [tex]\(f(x) = \sqrt{x}\)[/tex].
- The new function is [tex]\(g(x) = \sqrt{x} - 4\)[/tex].
2. Comparing [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
- Notice that [tex]\(g(x)\)[/tex] takes the form of [tex]\(f(x)\)[/tex] but with a subtraction of 4. Specifically, [tex]\(g(x) = f(x) - 4\)[/tex].
3. Interpreting the Transformation:
- The term [tex]\(- 4\)[/tex] in [tex]\(g(x) = \sqrt{x} - 4\)[/tex] indicates a vertical shift.
- A subtraction of a constant [tex]\(k\)[/tex] from a function [tex]\(f(x)\)[/tex] translates the graph downward by [tex]\(k\)[/tex] units.
- Therefore, subtracting 4 from [tex]\(\sqrt{x}\)[/tex] shifts the entire graph of [tex]\(\sqrt{x}\)[/tex] downward by 4 units.
4. Concluding the Transformation:
- The effect of the [tex]\(-4\)[/tex] is to move each point on the graph of [tex]\(f(x)\)[/tex] vertically downward by 4 units.
Thus, the correct interpretation of the transformation is:
D) Translates the graph 4 units downward.
