Linear Function
A linear function is a function whose graph is a straight line, described by an equation of the form f(x) = mx + b.
Formula
f(x) = mx + b
Definition
A linear function makes a straight line when graphed, with the form $f(x) = mx + b$, where $m$ is the slope (the rate of change, or steepness) and $b$ is the $y$-intercept where the line crosses the $y$-axis; the degree of $x$ is $1$, and the slope $m = (y_2 - y_1) / (x_2 - x_1)$ between any two points. Strictly speaking, in linear algebra a linear map must satisfy $f(ax + by) = af(x) + bf(y)$, which requires $b = 0$; functions with $b \neq 0$ are technically affine, and the distinction matters in higher mathematics.
Example
$f(x) = 2x + 3$ rises $2$ units for every $1$ unit it moves right and crosses the $y$-axis at $(0, 3)$, passing through points like $(0, 3)$, $(1, 5)$, $(2, 7)$. A cell phone plan charging $\$25$/month plus $\$0.10$ per text, $\text{Cost} = 0.10x + 25$, is linear since each additional text costs the same $\$0.10$: after $100$ texts, the cost is $\$25 + \$10 = \$35$. In linear algebra, $T(x) = Ax$ for a matrix $A$ is a true linear transformation; the slope-intercept form is affine, while the origin-passing form $y = mx$ is the linear map in the strict algebraic sense.
Key Insight
Linear means "makes a line": the rate of change is constant, every time $x$ increases by $1$, $y$ increases by the same amount, the slope, with no curves allowed, and any two points determine a unique line. Linear functions form the foundation of calculus: derivatives approximate functions locally by linear functions (tangent lines), and this linear approximation underlies Newton's method and numerical analysis.