Slope
In short: Slope is the number that tells how steep a line is: the vertical change divided by the horizontal change between any two points on that line.
8th GradeFunctions and Graphs8.EE.B.5
What is Slope?
The slope of a line measures its steepness and its direction. A line that climbs from left to right has a positive value, a line that falls has a negative one, and a flat line has a value of zero.
Precisely, for two points \((x_1, y_1)\) and \((x_2, y_2)\) with \(x_1 \neq x_2\), the steepness is the rise over the run, \(m = \frac{y_2 – y_1}{x_2 – x_1}\). The same value appears whichever two points of the line you pick.
How to spot it
- A word problem gives a rate such as dollars per hour, feet per second or gallons per mile.
- You are asked how steep a ramp, roof or road is, or which of two lines is steeper.
- A graph shows a straight line and the question asks how much y changes when x goes up by 1.
- An equation is written as y = mx + b and the question focuses on the number in front of x.
The method
- Pick two points on the line and write their coordinates carefully, keeping the same order for both.
- Compute the rise \(y_2 – y_1\) and the run \(x_2 – x_1\), watching the signs.
- Divide the rise by the run and simplify the fraction.
- Check the sign against the graph: rising means positive, falling means negative.
Slope: solved example
Problem
A ski trail at a Vermont resort drops from an elevation of 3,140 feet to 2,465 feet over a horizontal distance of 4,500 feet. Find the slope of the trail.
Solution
Rise: \(2{,}465 – 3{,}140 = -675\) feet, since the trail goes down. Run: \(4{,}500\) feet.
Divide: \(m = \frac{-675}{4{,}500} = -0.15\). The trail loses 0.15 foot of height for every foot traveled forward, so its slope is \(-0.15\).
Try it
Question 1
Find the slope of the line through \((-2, 9)\) and \((4, -3)\).
Show the answer
\(\frac{-3 – 9}{4 – (-2)} = \frac{-12}{6} = -2\).
Question 2
A line passes through \((5, 8)\) and \((5, 1)\). What can you say about its steepness?
Show the answer
The run is \(5 – 5 = 0\), so the division is impossible: a vertical line has an undefined slope.
Slope vs. Y-Intercept
The slope says how fast y changes as x grows; it is the m in \(y = mx + b\). The y-intercept says where the line starts on the vertical axis; it is the b. Two parallel lines share the same m but have different values of b.
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Whole dictionary: A–Z · 8th Grade · Functions and Graphs