Wren recorded an outside temperature of $-2^{\circ} F$ at 8 a.m. When she checked the temperature again, it was $4^{\circ} F$ at 12:00 p.m. If $x$ represents the time and $y$ represents the temperature in degrees Fahrenheit, what is the slope of the line through these two data points?
A. -1.5
B. -0.5
C. 0.5
D. 1.5
Answer (1)
Identify the two data points: (8, -2) and (12, 4).
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the values: m = 12 − 8 4 − ( − 2 ) = 4 6 .
Simplify to find the slope: m = 1.5 .
Explanation
Understanding the Problem We are given two data points representing the temperature at different times. The first point is (8 a.m., -2°F) and the second point is (12 p.m., 4°F). We need to find the slope of the line that passes through these two points.
Slope Formula The slope of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1
Substituting the Values In our case, we have ( x 1 , y 1 ) = ( 8 , − 2 ) and ( x 2 , y 2 ) = ( 12 , 4 ) . Substituting these values into the slope formula, we get: m = 12 − 8 4 − ( − 2 ) = 12 − 8 4 + 2 = 4 6
Simplifying the Slope Simplifying the fraction, we have: m = 4 6 = 2 3 = 1.5 Therefore, the slope of the line through these two data points is 1.5.
Examples
Understanding the slope can help us predict how the temperature changes over time. For example, if we know the temperature at two different times, we can calculate the slope and use it to estimate the temperature at other times. This is useful in many real-world applications, such as weather forecasting and climate modeling. The slope tells us the rate of change, in this case, how many degrees Fahrenheit the temperature changes per hour. Knowing this rate, we can make informed predictions about future temperatures, assuming the rate remains constant.
