The slope formula is ( m = frac{y_2 – y_1}{x_2 – x_1} ) (rise over run). A line with slope m through ( (x_1, y_1) ) has point-slope form ( y – y_1 = m(x – x_1) ) and slope-intercept form ( y = mx + b ), where b is the y-intercept. For example: through ( (1, 2) ) and ( (4, 11) ), ( m = 3 ) and the line is ( y = 3x – 1 ).
The slope of a line measures how steep it is: how much y changes for each one-unit change in x. Through two points ( (x_1, y_1) ) and ( (x_2, y_2) ), the slope formula is
$$m = frac{y_2 - y_1}{x_2 - x_1} = frac{text{rise}}{text{run}}$$
A slope of 3 means the line goes up 3 units for every 1 unit to the right. However, a slope of ( -frac12 ) means it goes down 1 unit for every 2 units to the right.
The four kinds of slope
| Slope | Direction | Example |
|---|---|---|
| Positive | rises left to right | ( y = 2x + 1 ) |
| Negative | falls left to right | ( y = -x + 4 ) |
| Zero | horizontal | ( y = 3 ) |
| Undefined | vertical | ( x = -2 ) |
How to find the slope of a line
From two points
Example 1. Find the slope through ( (1, 2) ) and ( (4, 11) ).
$$m = frac{11 - 2}{4 - 1} = frac{9}{3} = 3$$
Example 2. Through ( (-2, 3) ) and ( (4, 0) ):
$$m = frac{0 - 3}{4 - (-2)} = frac{-3}{6} = -frac12$$
It doesn’t matter which point you call first, as long as you use the same order on top and bottom.
From a graph
Pick two points where the line crosses grid corners exactly. Count the rise (up is positive, down is negative) and the run (always count to the right), then divide.
From an equation
Solve for y. In ( 4x + 2y = 10 ), ( y = -2x + 5 ), so the slope is ( -2 ). In standard form ( Ax + By = C ), the slope is ( -frac{A}{B} ).
Slope-intercept form
$$y = mx + b$$
Here m is the slope and b is the y-intercept, where the line crosses the y-axis. It’s sometimes searched as “slope intersection form”, but the standard name is slope-intercept form. It’s the easiest form to graph: plot ( (0, b) ), then use the slope to step to a second point.
Example 3. Graph ( y = frac23x – 1 ): start at ( (0, -1) ), then go up 2 and right 3 to ( (3, 1) ), and draw the line.
Point-slope form
When you know one point and the slope, use
$$y - y_1 = m(x - x_1)$$
Example 4. Find the line through ( (1, 2) ) and ( (4, 11) ). The slope is 3 (Example 1). Using the first point:
$$y - 2 = 3(x - 1)$$
Distribute and solve for y to convert to slope-intercept form: ( y = 3x – 1 ). In standard form that’s ( 3x – y = 1 ).
Example 5. Find the line with slope ( -frac12 ) through ( (-2, 3) ): ( y – 3 = -frac12(x + 2) ), which simplifies to ( y = -frac12x + 2 ).
Slope calculator
Enter two points to get the slope as an exact fraction and the line in all three forms.
The full-page slope calculator has more examples.
Parallel and perpendicular lines
- Parallel lines have equal slopes.
- Perpendicular lines have slopes that are negative reciprocals: ( m_1 m_2 = -1 ).
Example 6. Find the line through ( (2, 5) ) perpendicular to ( y = 4x – 7 ). The perpendicular slope is ( -frac14 ), so
$$y - 5 = -frac14(x - 2) quadRightarrowquad y = -frac14x + frac{11}{2}$$
Slope as a rate of change
Slope has units: “units of y per unit of x”. For example, if a taxi’s fare is ( C = 2.5d + 4 ) dollars for d miles, the slope 2.5 means $2.50 per mile. The intercept 4 is the fixed starting fee. Also, if a car is 120 km from home at 1 pm and 300 km at 3 pm, the slope of distance against time is ( frac{180}{2} = 90 ) km/h, its average speed.
x- and y-intercepts
The y-intercept is where the line crosses the y-axis (set ( x = 0 )); the x-intercept is where it crosses the x-axis (set ( y = 0 )).
Example. For ( 3x + 2y = 12 ): with ( x = 0 ), ( y = 6 ), so the y-intercept is ( (0, 6) ); with ( y = 0 ), ( x = 4 ), so the x-intercept is ( (4, 0) ). Plotting the two intercepts is the fastest way to graph a line in standard form. The slope is ( -frac{6}{4} = -frac32 ).
Average rate of change
For any function, the average rate of change from ( x = a ) to ( x = b ) is the slope of the line through ( (a, f(a)) ) and ( (b, f(b)) ):
$$text{average rate of change} = frac{f(b) - f(a)}{b - a}$$
Example. For ( f(x) = x^2 ) from ( x = 1 ) to ( x = 4 ): ( frac{16 – 1}{4 – 1} = 5 ). If x is time in seconds and ( f(x) ) is distance in metres, that’s an average speed of 5 m/s (see velocity and acceleration).
From slope to derivatives
A straight line has the same slope everywhere. However, a curve doesn’t. Calculus handles this by zooming in: the slope between two points on a curve (a secant line) gets closer and closer to a single value as the points move together. That limit is the slope of the tangent line, and it’s called the derivative:
$$f'(a) = lim_{hto0}frac{f(a + h) - f(a)}{h}$$
The fraction is just the slope formula with points ( (a, f(a)) ) and ( (a + h, f(a + h)) ).
Example 7. For ( f(x) = x^2 ) at ( x = 3 ), the derivative is ( f'(3) = 6 ), so the tangent line is ( y – 9 = 6(x – 3) ), point-slope form again. See the equation of a tangent line and the tangent line calculator.
Common mistakes
- Run over rise. Slope is change in y over change in x, not the other way round.
- Mixed order. ( frac{y_2 – y_1}{x_1 – x_2} ) gives the wrong sign.
- Subtracting negatives. ( 4 – (-2) = 6 ).
- Zero vs undefined. Horizontal lines have slope 0. However, vertical lines have no slope at all.
- Reading b from point-slope form. In ( y – 2 = 3(x – 1) ), the intercept is not 2. Instead, it’s ( -1 ) after simplifying.
Practice problems
- Find the slope through ( (3, 7) ) and ( (5, 15) ).
- Find the slope through ( (-1, 4) ) and ( (3, -4) ).
- Find the slope of ( 6x – 3y = 12 ).
- Write the line with slope 5 through ( (2, -1) ) in slope-intercept form.
- Find the line through ( (0, 3) ) parallel to ( y = -2x + 8 ).
- What is the slope of a line perpendicular to ( y = frac35x + 1 )?
Answers: (1) 4; (2) ( -2 ); (3) 2; (4) ( y = 5x – 11 ); (5) ( y = -2x + 3 ); (6) ( -frac53 ).
FAQ
What is the slope formula?
( m = frac{y_2 – y_1}{x_2 – x_1} ), the change in y divided by the change in x between two points.
What is point-slope form?
( y – y_1 = m(x – x_1) ), the equation of a line through ( (x_1, y_1) ) with slope m.
What is slope-intercept form?
( y = mx + b ), where m is the slope and b is the y-intercept.
How do you find the slope of a line from an equation?
Rearrange it into ( y = mx + b ); the coefficient of x is the slope.
What does a negative slope mean?
The line falls as you move from left to right: y decreases as x increases.
Further reading
- Lines (Paul’s Online Math Notes) — slope, the forms of a line, and parallel and perpendicular lines.
- Slope (Wikipedia) — definitions, notation and the link to derivatives.
Calculators for this topic
Keep learning
The Unit Circle: Angles, Coordinates and Trig Values
The unit circle explained: why the coordinates are cos and sin, every special angle in degrees and radians, tangent on the unit circle, and memory tricks.
Trigonometric Functions: Sin, Cos, Tan, Table and Graphs
The six trig functions with SOH-CAH-TOA and the unit circle, a table of exact values (sin 30, cos 180, sec 45), graphs, period and amplitude.