To find a linear function that models the data given in the table, we need to determine the slope (m) and the intercept (b) of the linear equation in the form [tex]\( f(x) = mx + b \)[/tex].
Given the data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & -6 \\
\hline
-1 & -1 \\
\hline
0 & 1 \\
\hline
2 & 4 \\
\hline
3 & 7 \\
\hline
\end{array}
\][/tex]
1. Calculate the slope (m):
The slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
However, since we need the best fit line through multiple points, we use a method that averages out the differences for all points. Without delving into the manual calculations (like calculating the covariance of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] and the variance of [tex]\(x\)[/tex]), we can state the computed slope from the aggregated calculations.
[tex]\[
m \approx 1.800
\][/tex]
2. Calculate the intercept (b):
Once the slope is determined, we use the formula for the intercept:
[tex]\[
b = \bar{y} - m\bar{x}
\][/tex]
where [tex]\(\bar{y}\)[/tex] is the mean of the [tex]\(y\)[/tex]-values and [tex]\(\bar{x}\)[/tex] is the mean of the [tex]\(x\)[/tex]-values.
Similarly, summarizing the calculations:
[tex]\[
b \approx 1.000
\][/tex]
3. Construct the linear equation:
Now that we have both the slope and intercept, we can construct our linear function.
[tex]\[
f(x) = 1.800x + 1.000
\][/tex]
Therefore, the linear function that models the data in the table is:
[tex]\[
f(x) = 1.800x + 1.000
\][/tex]
