Rise
Rise measures the change in the y-values between two points.
Free Algebra Guide
Understanding Slope
Learn how to find slope from two points, a graph, or a linear equation, understand rise over run, and recognize positive, negative, zero, and undefined slope.
Slope describes how a line changes as you move across a coordinate plane. It tells you both the direction of the line and how quickly the y-value changes compared with the x-value. You can find slope from coordinate points, from a graph, or directly from certain forms of a linear equation.
What Is Slope?
Slope is a number that describes how much a line changes vertically compared with how much it changes horizontally.
Imagine moving from one point on a line to another. If you move to the right, the line may rise, fall, or stay level. Slope measures that relationship.
Run
Run measures the change in the x-values between the same two points.
A Simple Way to Think About Slope
Slope answers the question: “How much does y change when x changes?”
The Slope Formula
When you know two points on a line, you can calculate the slope using the coordinates of those points.
Slope Formula
m =
y₂ − y₁
x₂ − x₁
The letter m represents slope. The numerator finds the change in y, and the denominator finds the change in x.
1. Label the points
Call one point
(x₁, y₁)
and the other
(x₂, y₂).
2. Subtract the y-values
Calculate
y₂ − y₁.
This is the rise.
3. Subtract the x-values
Calculate
x₂ − x₁.
This is the run.
4. Divide
Divide the change in y by the change
in x and simplify the result.
Keep the Subtraction Order Consistent
If you calculate y₂ − y₁ in the numerator, you must also calculate x₂ − x₁ in the denominator. Reversing both is fine. Reversing only one changes the sign of the answer.
What Does Rise Over Run Mean?
The slope formula and the phrase rise over run describe the same idea. Rise is the vertical change, and run is the horizontal change.
Consider the two points (1, 2) and (4, 8).
Rise
8 − 2 = 6
Run
4 − 1 = 3
Slope
6 / 3 = 2
A slope of 2 means the line rises 2 units for
every 1 unit it moves to the right.
How to Find Slope From Two Points
Suppose a line passes through (2, 3) and (6, 11).
Substitute Into the Formula
m =
11 − 3
6 − 2
=
8
4
= 2
1. Find the rise
11 − 3 = 8
2. Find the run
6 − 2 = 4
3. Divide
8 ÷ 4 = 2
4. Interpret
The line rises 2 units for every
1 unit it moves right.
How to Find Slope From a Graph
When a line is already graphed, you do not necessarily need to begin by writing down all four coordinate values. Instead, choose two convenient points on the line and count the vertical and horizontal movement between them.
1. Choose two points
Use clear grid intersections that lie
exactly on the line.
2. Count the rise
Move vertically from the first point
until you reach the y-level of the
second point.
3. Count the run
Move horizontally until you reach the
second point.
4. Divide rise by run
Keep track of direction so the sign of
the slope is correct.
Why the Graph and Formula Agree
Counting rise and run on a graph is simply a visual version of subtracting the y-coordinates and x-coordinates in the slope formula.
How to Find Slope From an Equation
The easiest linear equations for identifying slope are written in slope-intercept form.
Slope-Intercept Form
y = mx + b
In this form:
Slope
The coefficient multiplying x tells you the slope.
Y-Intercept
The constant tells you where the line crosses the y-axis.
y = 3x + 5
slope = 3
slope = 3
y = −
2
3
x + 4
slope = −2/3
slope = −2/3
Invisible Coefficients Still Count
In y = x + 4, the coefficient of x is 1, so the slope is 1.
In y = −x + 4, the coefficient is −1, so the slope is −1.
Positive, Negative, Zero, and Undefined Slope
You can often identify the type of slope simply by looking at the direction of the line from left to right.
The line rises from left to right.
The line falls from left to right.
The line is horizontal.
The line is vertical.
| Type | Direction | Rise / Run | Example |
|---|---|---|---|
| Positive | Rises left to right | Positive ratio | m = 2 |
| Negative | Falls left to right | Negative ratio | m = −3/4 |
| Zero | Horizontal | 0 / nonzero | m = 0 |
| Undefined | Vertical | nonzero / 0 | Undefined |
Zero and Undefined Are Not the Same
A horizontal line has no vertical change: its rise is 0. Therefore its slope is 0.
A vertical line has no horizontal change: its run is 0. Dividing by zero is undefined, so a vertical line has undefined slope.
Fractional and Negative Slopes
Slope does not have to be a whole number. A fractional slope simply means the vertical and horizontal changes do not simplify to a denominator of 1.
m =
2
5
One interpretation is: rise 2 units and run 5 units.
m = −
3
4
Moving to the right 4 units causes the line to fall 3 units.
A negative sign tells you the direction of
change. It does not mean you calculated the
slope incorrectly.
Worked Slope Examples
(−2, 1) and (4, 7)
- Rise: 7 − 1 = 6.
- Run: 4 − (−2) = 6.
- m = 6/6.
- m = 1.
(1, 6) and (5, 2)
- Rise: 2 − 6 = −4.
- Run: 5 − 1 = 4.
- m = −4/4.
- m = −1.
(0, 2) and (6, 5)
- Rise: 5 − 2 = 3.
- Run: 6 − 0 = 6.
- m = 3/6.
- m = 1/2.
y = −4x + 9
- The equation is in y = mx + b form.
- The coefficient of x is −4.
- Therefore m = −4.
- Slope = −4.
(−3, 5) and (4, 5)
- Rise: 5 − 5 = 0.
- Run: 4 − (−3) = 7.
- m = 0/7.
- m = 0.
(3, −2) and (3, 6)
- Rise: 6 − (−2) = 8.
- Run: 3 − 3 = 0.
- m = 8/0.
- Slope is undefined.
Common Slope Mistakes
Using Run Over Rise
Slope is rise over run, not run over rise. The change in y belongs in the numerator.
Reversing Only One Subtraction
If you calculate y₂ − y₁, you must also calculate x₂ − x₁. Changing the order in only one part incorrectly changes the sign.
Forgetting That Subtracting a Negative Adds
For example, 4 − (−2) = 6, not 2. Parentheses help prevent sign mistakes.
Calling a Vertical Line Slope Zero
Vertical lines have undefined slope because their run is 0.
Missing an Invisible 1
In y = x + 3, the slope is 1. In y = −x + 3, the slope is −1.
Automatically Turning Every Fraction Into a Decimal
Fractions often make rise and run easier to interpret. A slope of 2/3 clearly means rise 2 for run 3.
A Reliable Strategy for Finding Slope
1. Identify the representation
Are you given two points, a graph,
or an equation?
2. Find the vertical change
Use the y-values or count the rise on
the graph.
3. Find the horizontal change
Use the x-values or count the run on
the graph.
4. Divide rise by run
Simplify the fraction if possible.
5. Check the sign
A rising line should have positive
slope. A falling line should have
negative slope.
6. Watch for special cases
Horizontal lines have slope 0.
Vertical lines have undefined slope.
Always ask whether your numerical answer
matches the direction of the line. This quick
visual check catches many sign errors.
Practice Finding Slope
Practice slope from two points, graphs, rise and run, and linear equations. The interactive tool also includes positive, negative, zero, and undefined slope classification with step-by-step feedback.
Open the Slope Practice ToolRelated Math Resources
Personalized Math Support
Need Help Making Sense of the Math?
If a concept still feels confusing, you do not have to work through it alone. Tell me what you are studying and where you are getting stuck, and I will help you determine the best next step.