What Is a Linear Function?
The linear function (or first-degree function) has the form f(x) = ax + b, where a is the slope (rate of change) and b is the y-intercept (initial value). Its graph is always a straight line.
f(x) = ax + b a > 0 → increasing function (line rises from left to right) a < 0 → decreasing function (line falls from left to right) a = 0 → constant function (horizontal line) Zero of the function: x₀ = −b/a (where f(x₀) = 0)
How to find the function from two points
Given (x₁, y₁) and (x₂, y₂): a = (y₂ − y₁) ÷ (x₂ − x₁) (slope) b = y₁ − a × x₁ (y-intercept) Example: points (1, 5) and (3, 11) a = (11 − 5) ÷ (3 − 1) = 3 b = 5 − 3(1) = 2 f(x) = 3x + 2
Frequently asked questions
What is the difference between a linear function and a proportional function?
A proportional function (strictly linear) is f(x) = ax (b = 0, passes through the origin). A linear function in the general sense includes any f(x) = ax + b. Many textbooks use the terms interchangeably.
Does the zero of a linear function always exist?
Yes, as long as a ≠ 0. The zero is always x₀ = −b/a. If a = 0 (constant), there is no zero (unless b = 0 as well).
How do you find where two lines cross?
Set the two functions equal: a₁x + b₁ = a₂x + b₂. Solve for x. That is the intersection point — fundamental for systems of linear equations.
Do linear functions have a maximum or minimum?
No! Since the line extends infinitely, there is no maximum or minimum value. This is different from the quadratic function, which has a vertex.