To determine the equation of the linear function that best fits the given data points [tex]\((x, y)\)[/tex], we can follow these steps:
1. Collect the Given Data Points:
From the table, we have the following pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
[tex]\[
\begin{array}{cccc}
(1, -6) & (3, 6) & (5, 18) & (7, 30) \\
\end{array}
\][/tex]
2. Calculate the Slope and Intercept:
A linear function is of the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the intercept.
Using linear regression techniques:
[tex]\[
m = \frac{(n \sum xy - \sum x \sum y)}{(n \sum x^2 - (\sum x)^2)} \\
b = \frac{(\sum y \sum x^2 - \sum x \sum xy)}{(n \sum x^2 - (\sum x)^2)}
\][/tex]
where [tex]\(n\)[/tex] is the number of data points.
Given:
[tex]\[
\begin{align*}
\sum x &= 1 + 3 + 5 + 7 = 16, \\
\sum y &= -6 + 6 + 18 + 30 = 48, \\
\sum xy &= 1(-6) + 3(6) + 5(18) + 7(30) = -6 + 18 + 90 + 210 = 312, \\
\sum x^2 &= 1^2 + 3^2 + 5^2 + 7^2 = 1 + 9 + 25 + 49 = 84.
\end{align*}
\][/tex]
Plugging these into the formulas:
[tex]\[
\begin{align*}
m &= \frac{4 \cdot 312 - 16 \cdot 48}{4 \cdot 84 - 16^2} \\
&= \frac{1248 - 768}{336 - 256} \\
&= \frac{480}{80} \\
&= 6, \\
b &= \frac{48 \cdot 84 - 16 \cdot 312}{4 \cdot 84 - 16^2} \\
&= \frac{4032 - 4992}{336 - 256} \\
&= \frac{-960}{80} \\
&= -12.
\end{align*}
\][/tex]
3. Form the Linear Equation:
With the calculated slope [tex]\(m = 6\)[/tex] and intercept [tex]\(b = -12\)[/tex], the equation of the linear function is:
[tex]\[
y = 6x - 12.
\][/tex]
Therefore, the correct equation that models the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] as given in the table is:
[tex]\[
\boxed{y = 6x - 12}
\][/tex]
