In the table below, [tex]\( x \)[/tex] represents the miles traveled and [tex]\( y \)[/tex] represents the cost to travel by train.

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

What is the slope of this function?

A. 0.44
B. 0.63
C. 2.25
D. 22.50



Answer :

To solve for the slope of the function given the points in the table, we must use the formula for the slope of a line between two points, which is given by:

[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Where [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are two points on the line. Let's choose the points furthest apart from the table to ensure we capture the overall rate of change accurately:

Using the points [tex]\((2, 8.50)\)[/tex] and [tex]\((12, 31.00)\)[/tex]:

1. Calculate the change in miles ([tex]\(\Delta x\)[/tex]):
[tex]\[ \Delta x = x_2 - x_1 = 12 - 2 = 10 \][/tex]

2. Calculate the change in cost ([tex]\(\Delta y\)[/tex]):
[tex]\[ \Delta y = y_2 - y_1 = 31.00 - 8.50 = 22.50 \][/tex]

3. Now, substitute [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the slope formula:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} = \frac{22.50}{10} = 2.25 \][/tex]

So, the slope of the function is:

[tex]\[ 2.25 \][/tex]

Hence, the correct answer is:

[tex]\[ \boxed{2.25} \][/tex]