To understand the effect of the [tex]\(-4\)[/tex] in the function [tex]\(g(x) = \sqrt{x} - 4\)[/tex] on the graph of [tex]\(f(x) = \sqrt{x}\)[/tex], let's examine the transformation step by step.
1. Identifying the Basic Function:
- The basic function given is [tex]\(f(x) = \sqrt{x}\)[/tex]. This function represents the square root graph, which starts at the origin [tex]\((0, 0)\)[/tex] and curves upward to the right.
2. Understanding the Transformation:
- The function [tex]\(g(x) = \sqrt{x} - 4\)[/tex] involves subtracting 4 from the value of [tex]\(f(x)\)[/tex]. This subtraction occurs outside of the square root function.
- A transformation outside the function [tex]\( f(x) = \sqrt{x} \)[/tex], specifically of the form [tex]\(f(x) - c\)[/tex], where [tex]\(c\)[/tex] is a constant, translates the graph vertically.
3. Determining the Direction:
- When we subtract a constant value [tex]\(c\)[/tex] from a function [tex]\(f(x)\)[/tex], it translates the graph of [tex]\(f(x)\)[/tex] downward by [tex]\(c\)[/tex] units. In this case, the constant [tex]\(c\)[/tex] is 4.
- Therefore, [tex]\(g(x) = \sqrt{x} - 4\)[/tex] will shift the graph of [tex]\(f(x) = \sqrt{x}\)[/tex] downward by 4 units.
4. Comparison with the Options:
- Option A) Translates the graph to the right by 4 units
- Option B) Translates the graph 4 units downward
- Option C) Translates the graph to the left by 4 units
- Option D) Translates the graph 4 units upward
From the explanation, we can see that the correct transformation for [tex]\(g(x) = \sqrt{x} - 4\)[/tex] is a downward shift of 4 units.
Thus, the answer is:
B) Translates the graph 4 units downward.
