linear function

Also called: straight line, y = a + bx

A relation of the form y = a + b·x: the intercept a is the value at x = 0 and the slope b is the change in y for each unit of x. Profit = c·n − F, a trend line and a tax with a fixed part are all linear functions.

A linear function is y = a + b·x. Two numbers fix it: the intercept a, where the line starts at x = 0, and the slope b, how much y changes when x grows by one unit.

$y = a + b \cdot x$

It crosses zero at x = −a ÷ b, which is how break-even falls out of profit = c·n − F. Two lines a₁ + b₁x and a₂ + b₂x cross at x = (a₂ − a₁) ÷ (b₁ − b₂), the point where one option starts to beat the other.

Only a line through zero, y = b·x, is a proportion: double x and y doubles. With an intercept, the ratio y ÷ x (an average cost, an effective tax rate) changes along the line.

Example

Klasna's monthly profit: profit = 174.48·n − 1,038,000.

Intercept: −1,038,000 (fixed costs with zero lessons); slope: +174.48 per lesson.
Zero at n = 1,038,000 ÷ 174.48 ≈ 5,949.1, so 5,950 lessons.
7,000 lessons give 183,360; 14,000 give 1,404,720, about 7.7 times more, not 2 times.

Common mistakes

Treating a line as a proportion. With an intercept, doubling x does not double y.
Constant effective rate. A fixed part makes y ÷ x fall as x grows (a tax weighs more on a small income).
Lines that are really staircases. Step costs such as ⌈n ÷ 400⌉ × salary jump; model them as steps.
Extending a line too far. Fixed costs stay fixed only within a range.

Learn it in the course

The linear function · Read y = a + bx in business formulas: profit = c·n − F, trend lines, a tax with a fixed part, the design effect, where two lines cross, and two relations that are not lines.
Pythagoras in statistics: distances and errors in quadrature · Euclidean and Manhattan distance, why independent errors add as squares, the pooled SE of a z-test vs the unpooled SE of an interval, and how to read the Wilson and Newcombe interval formulas.
Standardizing: z, CV, effect size and r · Put different scales side by side: z-scores, the coefficient of variation, t as signal over noise, Cohen's d as an effect in SD units, and correlation as a covariance scaled to between −1 and 1.
Slope as a marginal quantity · What one more unit adds: marginal vs average CAC and cost, operating leverage as 1 ÷ margin of safety, and price elasticity as a ratio of relative changes.