To determine how [tex]\( f(x) \)[/tex] was transformed to create [tex]\( g(x) \)[/tex], we need to analyze the corresponding [tex]\( y \)[/tex]-values for each [tex]\( x \)[/tex]-value in the given tables.
Let's set up the values:
For [tex]\( f(x) \)[/tex]:
- [tex]\( f(-2) = \frac{1}{9} \)[/tex]
- [tex]\( f(-1) = \frac{1}{3} \)[/tex]
- [tex]\( f(2) = 9 \)[/tex]
- [tex]\( f(3) = 27 \)[/tex]
- [tex]\( f(4) = 81 \)[/tex]
For [tex]\( g(x) \)[/tex]:
- [tex]\( g(-2) = -\frac{17}{9} \)[/tex]
- [tex]\( g(-1) = -\frac{5}{3} \)[/tex]
- [tex]\( g(2) = 7 \)[/tex]
- [tex]\( g(3) = 25 \)[/tex]
- [tex]\( g(4) = 79 \)[/tex]
Next, we calculate the differences between the corresponding [tex]\( y \)[/tex]-values of [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex]:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) - f(-2) = -\frac{17}{9} - \frac{1}{9} = -\frac{17 + 1}{9} = -\frac{18}{9} = -2
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) - f(-1) = -\frac{5}{3} - \frac{1}{3} = -\frac{5 + 1}{3} = -\frac{6}{3} = -2
\][/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) - f(2) = 7 - 9 = -2
\][/tex]
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) - f(3) = 25 - 27 = -2
\][/tex]
5. For [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) - f(4) = 79 - 81 = -2
\][/tex]
Observe that the difference between [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] is consistently [tex]\(-2\)[/tex] for all [tex]\( x \)[/tex]-values.
The consistent difference of [tex]\(-2\)[/tex] indicates that [tex]\( g(x) \)[/tex] was obtained by vertically shifting [tex]\( f(x) \)[/tex] downwards by 2 units.
Thus, the correct transformation is:
[tex]\[ \boxed{D. \text{Horizontal or vertical shift}}\][/tex]
