Linear Function
Functions & Advanced AlgebraA 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. Its equation has the form $f(x) = mx + b$, where $m$ is the slope (steepness) and $b$ is where the line crosses the $y$-axis.
Example
$f(x) = 2x + 3$: the line rises $2$ units for every $1$ unit it moves right, and crosses the $y$-axis at $(0, 3)$. It's a straight line through points like $(0, 3)$, $(1, 5)$, $(2, 7)$.
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). No curves allowed.
Definition
A linear function has the form $f(x) = mx + b$, where $m$ is the slope (rate of change) and $b$ is the $y$-intercept. The degree of $x$ is $1$. The slope $m = (y_2 - y_1) / (x_2 - x_1)$ between any two points.
Example
A cell phone plan charges $\$25$/month plus $\$0.10$ per text. $\text{Cost} = 0.10x + 25$. This is linear: each additional text costs $\$0.10$. After $100$ texts: $\$25 + \$10 = \$35$.
Key Insight
Linear functions model constant rates of change. Any two points determine a unique line. The slope $m$ represents units of output per unit of input (a rate).
Definition
A linear function $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(x) = mx + b$. Note: strictly speaking, linear maps in linear algebra satisfy $f(ax + by) = af(x) + bf(y)$, which requires $b = 0$. Functions with $b \neq 0$ are affine. The distinction matters in higher mathematics.
Example
In linear algebra, $T(x) = Ax$ for a matrix $A$ is a linear transformation. The slope-intercept form is affine; the origin-passing form $y = mx$ is the true linear map in the algebraic sense.
Key Insight
Linear functions form the foundation of calculus: derivatives approximate functions locally by linear functions (tangent lines). Linear approximation underlies Newton's method and numerical analysis.