Given the quadratic functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we need to determine how [tex]\( f(x) \)[/tex] should be shifted to match [tex]\( g(x) \)[/tex].
First, let's analyze the given values in the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
-6 & 36 & 4 \\
-5 & 25 & 1 \\
-4 & 16 & 0 \\
-3 & 9 & 1 \\
-2 & 4 & 4 \\
-1 & 1 & 9 \\
0 & 0 & 16 \\
1 & 1 & 25 \\
2 & 4 & 36 \\
\hline
\end{array}
\][/tex]
We can see that [tex]\( f(x) \)[/tex] is a classic quadratic function [tex]\( f(x) = x^2 \)[/tex].
Now let's track the relationship between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] by examining corresponding values:
- When [tex]\( x = -6 \)[/tex], [tex]\( f(-6) = 36 \)[/tex] and [tex]\( g(x) = 36 \)[/tex] when [tex]\( x = 2 \)[/tex].
- When [tex]\( x = -5 \)[/tex], [tex]\( f(-5) = 25 \)[/tex] and [tex]\( g(x) = 25 \)[/tex] when [tex]\( x = 1 \)[/tex].
- When [tex]\( x = -4 \)[/tex], [tex]\( f(-4) = 16 \)[/tex] and [tex]\( g(x) = 16 \)[/tex] when [tex]\( x = 0 \)[/tex].
- When [tex]\( x = -2 \)[/tex], [tex]\( f(-2) = 4 \)[/tex] and [tex]\( g(x) = 4 \)[/tex] when [tex]\( x = -2 \)[/tex].
From the analysis above, when we move the [tex]\( x \)[/tex] value from [tex]\( f(x) \)[/tex] by +4, we get the desired [tex]\( g(x) \)[/tex] value. This indicates a horizontal shift of 4 units to the right.
Thus, to match [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex], [tex]\( f(x) \)[/tex] must be shifted:
Right by 4 units
This change transforms [tex]\( f(x) = x^2 \)[/tex] into [tex]\( g(x) = (x-4)^2 \)[/tex], aligning [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex] accurately.
