To determine the slope of the line passing through the two data points, we will use the slope formula and follow a step-by-step process.
First, label the coordinates of the two data points. Let:
- [tex]\( (x_1, y_1) \)[/tex] be the point when the temperature was first recorded.
- [tex]\( (x_2, y_2) \)[/tex] be the point when the temperature was recorded again later.
From the problem, we know:
- The first recorded temperature is [tex]\( -2^{\circ} F \)[/tex] at 8 a.m., which translates to the point [tex]\((8, -2)\)[/tex].
- The second recorded temperature is [tex]\( 4^{\circ} F \)[/tex] at 12:00 p.m., which translates to the point [tex]\((12, 4)\)[/tex].
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the values from our coordinates:
[tex]\[ x_1 = 8, \; y_1 = -2, \; x_2 = 12, \; y_2 = 4 \][/tex]
Use the formula to find the slope:
[tex]\[ m = \frac{4 - (-2)}{12 - 8} \][/tex]
Simplify the numerator:
[tex]\[ 4 - (-2) = 4 + 2 = 6 \][/tex]
Simplify the denominator:
[tex]\[ 12 - 8 = 4 \][/tex]
So the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{6}{4} = 1.5 \][/tex]
Therefore, the slope of the line through these two data points is:
[tex]\[ \boxed{1.5} \][/tex]
