To solve this problem, we need to understand how the given function [tex]\( f(x) = \frac{100}{x} - 5 \)[/tex] is derived from the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex].
We start with the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], which is a hyperbola.
1. Vertical Stretch:
The first transformation in [tex]\( f(x) = \frac{100}{x} - 5 \)[/tex] is multiplying the reciprocal function by 100. This transforms the function [tex]\( f(x) = \frac{1}{x} \)[/tex] to [tex]\( f(x) = \frac{100}{x} \)[/tex]. Mathematically, multiplying by 100 is a vertical stretch:
[tex]\[
\frac{1}{x} \xrightarrow[]{\text{vertical stretch by 100}} \frac{100}{x}
\][/tex]
2. Vertical Translation:
The second transformation is the subtraction of 5. This translation moves every point on the graph of [tex]\( f(x) = \frac{100}{x} \)[/tex] down by 5 units:
[tex]\[
\frac{100}{x} \xrightarrow[]{\text{translate down by 5}} \frac{100}{x} - 5
\][/tex]
Combining these two transformations, we get the function [tex]\( f(x) = \frac{100}{x} - 5 \)[/tex].
Thus, the original graph [tex]\( f(x) = \frac{1}{x} \)[/tex] undergoes a vertical stretch by a factor of 100 and a vertical translation 5 units down to become the graph of [tex]\( f(x) = \frac{100}{x} - 5 \)[/tex].
So, the correct transformation description is:
"It is a vertical stretch with a factor of 100 and a translation 5 units down."
Therefore, the correct option is:
It is a vertical stretch with a factor of 100 and a translation 5 units down.
