The function [tex]$g(x)$[/tex] is a transformation of the parent function [tex]$f(x)$[/tex]. Decide how [tex]$f(x)$[/tex] was transformed to make [tex]$g(x)$[/tex].

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c}{[tex]$f(x)$[/tex]} \\
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline-2 & [tex]$\frac{1}{9}$[/tex] \\
\hline-1 & [tex]$\frac{1}{3}$[/tex] \\
\hline 2 & 9 \\
\hline 3 & 27 \\
\hline 4 & 81 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c}{[tex]$g(x)$[/tex]} \\
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline-2 & [tex]$-\frac{17}{9}$[/tex] \\
\hline-1 & [tex]$-\frac{5}{3}$[/tex] \\
\hline 2 & 7 \\
\hline 3 & 25 \\
\hline 4 & 79 \\
\hline
\end{tabular}

A. Reflection across the line [tex]$y=x$[/tex]
B. Horizontal or vertical stretch
C. Horizontal or vertical reflection
D. Horizontal or vertical shift



Answer :

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]