Slope Formula
In short: The slope formula computes steepness from two points, \(m = \frac{y_2 – y_1}{x_2 – x_1}\), by dividing the change in height by the change in horizontal position.
8th GradeFunctions and Graphs8.EE.B.6
What is Slope Formula?
The slope formula turns two coordinate pairs into one number. For points \((x_1, y_1)\) and \((x_2, y_2)\), it states \(m = \frac{y_2 – y_1}{x_2 – x_1}\), provided \(x_1 \neq x_2\).
Subtract the y-values on top and the x-values on the bottom, taking the points in the same order in both. Swapping which point is first flips both signs and leaves the result unchanged. A positive answer means the line climbs, a negative one means it falls, zero means horizontal, and a zero denominator means vertical.
How to spot it
- A problem lists two ordered pairs and asks how fast one coordinate changes relative to the other.
- A line is described by two points rather than by an equation, and its direction is needed.
- Two points share the same x-value or the same y-value and the result seems special.
- A table has several rows, and you must check that the ratio of changes is the same between each pair.
The method
- Label the points \((x_1, y_1)\) and \((x_2, y_2)\), choosing either one as first.
- Subtract in the numerator, \(y_2 – y_1\), and in the denominator, \(x_2 – x_1\), in the same order.
- Simplify the fraction and keep the sign.
- Interpret: positive rises, negative falls, zero is flat, and an undefined result is a vertical line.
Slope Formula: solved example
Problem
A delivery drone is at the point \((-4, 7)\) on a map grid, with height in tens of feet, and later at \((8, -2)\). Use the slope formula for the line through these two points.
Solution
Substitute: \(m = \frac{-2 – 7}{8 – (-4)} = \frac{-9}{12}\).
Divide top and bottom by 3: \(m = \) \(-\frac{3}{4}\). The line falls 3 units for every 4 units it moves to the right.
Try it
Question 1
Apply the slope formula to \((2, 5)\) and \((6, 17)\).
Show the answer
\(\frac{17 – 5}{6 – 2} = \frac{12}{4} = 3\).
Question 2
What does the formula give for \((-1, -6)\) and \((-1, 9)\)?
Show the answer
The denominator is \(-1 – (-1) = 0\), so the slope is undefined: the line is vertical.
Slope Formula vs. Rate of Change
The slope formula is a computation for two points on a straight line, using coordinates. Rate of change is the idea it measures: how much one quantity changes per unit of another, and it also applies to tables and graphs that are not straight. On a line the two numbers agree.
Keep going
Lessons
Worksheets and quizzes
- Slope and Similar Triangles – Worksheet
- Slope and Similar Triangles – Quiz
- Slope and Rate of Change – Worksheet
- Slope and Rate of Change – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Functions and Graphs