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\begin{tabular}{|c|c|}
\hline Miles, [tex]$x$[/tex] & Cost, [tex]$y$[/tex] \\
\hline 2 & 8.50 \\
\hline 5 & 15.25 \\
\hline 8 & 22.00 \\
\hline 12 & 31.00 \\
\hline
\end{tabular}

What is the [tex]$y$[/tex]-intercept of this function?

A. 2.25
B. 4.00
C. 6.50
D. 17.13


Sagot :

To determine the [tex]\( y \)[/tex]-intercept of a linear function given by the data points, let's walk through the process of performing a linear regression. This technique helps us find the line of best fit for the given data. The linear equation can be represented in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.

The provided data points are:
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 8.50 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 15.25 \)[/tex]
- When [tex]\( x = 8 \)[/tex], [tex]\( y = 22.00 \)[/tex]
- When [tex]\( x = 12 \)[/tex], [tex]\( y = 31.00 \)[/tex]

To find the equation of the line, we need to determine the slope ([tex]\( m \)[/tex]) and the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]):

1. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] of the line can be found using the formula:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
where [tex]\( n \)[/tex] is the number of data points.

2. Calculate the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]:
Once we have the slope, we can find the [tex]\( y \)[/tex]-intercept using the formula:
[tex]\[ b = \frac{\sum y - m (\sum x)}{n} \][/tex]

Given the calculations, we find:
- The slope, [tex]\( m \)[/tex], is 2.25.
- The [tex]\( y \)[/tex]-intercept, [tex]\( b \)[/tex], is 4.00.

Therefore, the [tex]\( y \)[/tex]-intercept of the linear function is:
[tex]\[ b = 4.00 \][/tex]

So, the correct answer is:
[tex]\[ \boxed{4.00} \][/tex]
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