To determine the transformation from the graph of the function [tex]\( f(x) = \frac{1}{x-2} + 1 \)[/tex] to the graph of the function [tex]\( g(x) = \frac{1}{x+5} + 9 \)[/tex], let's analyze the changes in detail step-by-step:
1. Analyze Horizontal Shifts (Changes in [tex]\( x \)[/tex]-terms):
- The denominator of [tex]\( f(x) \)[/tex] is [tex]\( x-2 \)[/tex]. This indicates a horizontal shift of 2 units to the right from the origin since it is in the form [tex]\( \frac{1}{x - h} \)[/tex].
- The denominator of [tex]\( g(x) \)[/tex] is [tex]\( x+5 \)[/tex]. This can be rewritten as [tex]\( x - (-5) \)[/tex], indicating a horizontal shift of 5 units to the left.
- Comparing these shifts:
- [tex]\( f(x) \)[/tex] is already shifted 2 units to the right.
- For [tex]\( g(x) \)[/tex], we need a total shift left of [tex]\( 5 + 2 = 7 \)[/tex] units to match its horizontal position.
2. Analyze Vertical Shifts (Changes in constants added outside the fraction):
- The constant term added in [tex]\( f(x) \)[/tex] is [tex]\( +1 \)[/tex].
- The constant term added in [tex]\( g(x) \)[/tex] is [tex]\( +9 \)[/tex].
- To go from [tex]\( +1 \)[/tex] in [tex]\( f(x) \)[/tex] to [tex]\( +9 \)[/tex] in [tex]\( g(x) \)[/tex], the graph needs to shift [tex]\( 9 - 1 = 8 \)[/tex] units upwards.
Thus, to transform the graph of [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex], the graph must shift [tex]\( 7 \)[/tex] units to the left and [tex]\( 8 \)[/tex] units up.
Therefore, the correct statement describing this transformation is:
- The graph shifts 7 units left and 8 units up.
