To understand the transformation of the graph of the function [tex]\( f \)[/tex] onto the graph of [tex]\( g \)[/tex], we need to analyze the changes in both the horizontal and vertical positions.
Given:
- [tex]\( f(x) = \frac{1}{x - 3} + 1 \)[/tex]
- [tex]\( g(x) = \frac{1}{x + 4} + 3 \)[/tex]
### Step-by-Step Transformation Analysis
1. Horizontal Shift Analysis:
- In [tex]\( f(x) \)[/tex], the denominator is [tex]\( x - 3 \)[/tex].
- In [tex]\( g(x) \)[/tex], the denominator is [tex]\( x + 4 \)[/tex].
To transform [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex], observe the shift in the [tex]\( x \)[/tex] term:
- [tex]\( f(x) \)[/tex]: [tex]\( x - 3 \)[/tex]
- [tex]\( g(x) \)[/tex]: [tex]\( x + 4 \)[/tex]
So, the term [tex]\( x - 3 \)[/tex] must be transformed to [tex]\( x + 4 \)[/tex]. This indicates a horizontal shift.
- [tex]\( (x - 3) \)[/tex] needs to move to the right by 7 units (since [tex]\(-3\)[/tex] must become [tex]\(+4\)[/tex], a shift of [tex]\(4 - (-3) = 7\)[/tex] units to the left).
2. Vertical Shift Analysis:
- In [tex]\( f(x) \)[/tex], the constant term is [tex]\( +1 \)[/tex].
- In [tex]\( g(x) \)[/tex], the constant term is [tex]\( +3 \)[/tex].
To transform the vertical position:
- [tex]\( f(x) \)[/tex]: constant is [tex]\( +1 \)[/tex]
- [tex]\( g(x) \)[/tex]: constant is [tex]\( +3 \)[/tex]
So, the constant [tex]\(1\)[/tex] must be transformed to [tex]\(3\)[/tex]. This indicates a vertical shift.
- [tex]\( 1 \)[/tex] needs to move up by 2 units (since [tex]\( 1 \)[/tex] must become [tex]\( 3 \)[/tex]).
### Conclusion:
The transformations required are:
- A horizontal shift of 7 units to the left.
- A vertical shift of 2 units up.
Therefore, the correct statement describing the transformation is:
"The graph shifts 7 units left and 2 units up."
Thus, the answer is:
[tex]\[ \boxed{\text{The graph shifts 7 units left and 2 units up.}} \][/tex]
