To determine the transformation needed to get [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex], let's analyze the given data and the relationship between the two functions:
The table for [tex]\( g(x) \)[/tex] shows the following values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-2 & -6 \\
\hline
-1 & -4 \\
\hline
0 & -2 \\
\hline
1 & 0 \\
\hline
2 & 2 \\
\hline
3 & 4 \\
\hline
4 & 6 \\
\hline
\end{array}
\][/tex]
From the table, we can observe that:
- For [tex]\( x = -2 \)[/tex], [tex]\( g(x) = -6 \)[/tex]
- For [tex]\( x = -1 \)[/tex], [tex]\( g(x) = -4 \)[/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( g(x) = -2 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( g(x) = 0 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( g(x) = 2 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( g(x) = 4 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( g(x) = 6 \)[/tex]
First, let's identify the linear function [tex]\( g(x) \)[/tex]:
The differences between the [tex]\( y \)[/tex]-values corresponding to increments of 1 in [tex]\( x \)[/tex] are consistent, indicating that [tex]\( g(x) \)[/tex] is a linear function with a constant slope.
The slope [tex]\( m \)[/tex] of [tex]\( g(x) \)[/tex] can be calculated using any two points from the table. Let's use the points [tex]\((-2, -6)\)[/tex] and [tex]\((-1, -4)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-6)}{-1 - (-2)} = \frac{-4 + 6}{-1 + 2} = \frac{2}{1} = 2
\][/tex]
Thus, [tex]\( g(x) \)[/tex] has a slope of 2.
Next, we determine the y-intercept [tex]\( b \)[/tex] of the function [tex]\( g(x) \)[/tex]. Using the point [tex]\( (x_2, y_2) = (-1, -4) \)[/tex]:
[tex]\[
g(x) = mx + b
\][/tex]
[tex]\[
-4 = 2(-1) + b
\][/tex]
[tex]\[
-4 = -2 + b
\][/tex]
[tex]\[
b = -2
\][/tex]
Thus, the equation for [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = 2x - 2
\][/tex]
Now, we need to compare this with [tex]\( f(x) \)[/tex]:
To find the transformation, we observe how the values of [tex]\( g(x) \)[/tex] change relative to [tex]\( f(x) \)[/tex]. From the given options:
1. [tex]\( g(x) = f(x-3) \)[/tex]: This represents a horizontal shift to the right by 3 units.
2. [tex]\( g(x) = f(x+3) \)[/tex]: This represents a horizontal shift to the left by 3 units.
3. [tex]\( g(x) = f(x) - 3 \)[/tex]: This represents a vertical shift downward by 3 units.
4. [tex]\( g(x) = f(x) + 3 \)[/tex]: This represents a vertical shift upward by 3 units.
From the table, it is evident that each value of [tex]\( g(x) \)[/tex] is 3 units smaller than the corresponding value of [tex]\( f(x) \)[/tex] at every [tex]\( x \)[/tex]:
Thus, the correct transformation is [tex]\( g(x) = f(x) - 3 \)[/tex].
The transformation of [tex]\( f(x) \)[/tex] that produces [tex]\( g(x) \)[/tex] is therefore:
[tex]\[
g(x) = f(x) - 3
\][/tex]
