What is slope and how do I calculate it?

Slope (m) measures how steep the line is. It equals "rise over run": m = (y₂ - y₁) / (x₂ - x₁). A slope of 2 means: for every 1 unit you move right, the line goes up 2 units.

What is the difference between slope-intercept and point-slope form?

Slope-intercept form (y = mx + b) is easiest when you know the slope and y-intercept. Point-slope form (y - y₁ = m(x - x₁)) is easiest when you know the slope and one specific point on the line.

Where are linear functions used in real life?

Linear functions appear everywhere: calculating the cost of items at a fixed price, speed-distance relationships (d = vt), converting temperatures (C = (F-32)×5/9), salary calculations, and predicting trends in data.

VIZMath · Function Guide

Linear Function

f(x) = mx + b

The straight line that models constant rate of change — speed, cost, temperature, and more.

Linear Function

A linear function is a polynomial function of degree 1, written as f(x) = mx + b. Its graph is always a straight line.

The coefficient m is called the slope — it tells you how steeply the line rises or falls. The constant b is the y-intercept — where the line crosses the y-axis.

If m > 0, the line goes upward from left to right; if m < 0, it goes downward. A linear function represents constant rate of change, making it one of the most fundamental functions in all of mathematics.

Standard Form

f(x) = mx + b

Slope — rise over run, rate of change (Δy / Δx)

Y-intercept — the value of f(x) when x = 0

Key Properties

Domain

All real numbers: (-∞, +∞)

Range

All real numbers: (-∞, +∞) (unless m = 0)

Slope

m = (y₂ - y₁) / (x₂ - x₁)

Y-intercept

(0, b)

X-intercept

(-b/m, 0) when m ≠ 0

Increasing/Decreasing

Increasing when m > 0; decreasing when m < 0; constant when m = 0

Examples

f(x) = 2x + 1

Slope 2 (rises steeply), y-intercept at (0, 1).

f(x) = -x + 3

Slope -1 (falls at 45°), y-intercept at (0, 3).

f(x) = 0.5x

Gentle upward slope through the origin.

Visualize the Linear Function

Drag the slope and intercept sliders to see the line update instantly

Try in VIZMath →

FAQ

A function is linear when the variable x appears only to the first power (no x², no 1/x). Its graph is always a perfectly straight line, and it shows a constant rate of change: equal changes in x always produce equal changes in y.

Related Functions

Quadratic Function Exponential Function