When examining the transformation from [tex]\( f(x) = \frac{1}{x} \)[/tex] to [tex]\( g(x) = \frac{1}{x} - 10 \)[/tex], we can determine how the graph of the function is affected by analyzing the changes in the function's formula.
1. Identify the transformation:
[tex]\( g(x) = \frac{1}{x} - 10 \)[/tex]
2. Understand vertical transformations:
- A vertical shift involves adding or subtracting a constant to the function [tex]\( f(x) \)[/tex].
- If a constant [tex]\( k \)[/tex] is subtracted from [tex]\( f(x) \)[/tex] to form [tex]\( g(x) = f(x) - k \)[/tex], the graph of [tex]\( f(x) \)[/tex] is shifted downward by [tex]\( k \)[/tex] units.
- If a constant [tex]\( k \)[/tex] is added to [tex]\( f(x) \)[/tex] to form [tex]\( g(x) = f(x) + k \)[/tex], the graph of [tex]\( f(x) \)[/tex] is shifted upward by [tex]\( k \)[/tex] units.
3. Analyze the specific transformation:
- Given [tex]\( g(x) = \frac{1}{x} - 10 \)[/tex], we see that [tex]\( 10 \)[/tex] is subtracted from [tex]\( f(x) \)[/tex].
4. Determine the shift:
- Since [tex]\( -10 \)[/tex] is subtracted from the original function [tex]\( f(x) \)[/tex], this results in a downward vertical shift.
5. Conclusion:
- The transformation causes the graph of [tex]\( f(x) = \frac{1}{x} \)[/tex] to shift 10 units downward.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{B. The graph of } f(x) \text{ is shifted 10 units down.}} \][/tex]
