What is a Linear Function?

A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.  It is linear: the exponent of the $x$ term is a one (first power), and it follows the definition of a function: for each input ($x$) there is exactly one output ($y$).  Also, its graph is a straight line.  

Graphs of Linear Functions

The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In the linear function graphs below, the constant, $m$, determines the slope or gradient of that line, and the constant term, $b$, determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept. 

Graphs of linear functions

The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.  The blue line has a positive slope of $\frac{1}{2}$ and a $y$-intercept of $-3$; the red line has a negative slope of $-1$ and a $y$-intercept of $5$.

Vertical and Horizontal Lines

Vertical lines have an undefined slope, and cannot be represented in the form $y=mx+b$, but instead as an equation of the form $x=c$ for a constant $c$, because the vertical line intersects a value on the $x$-axis, $c$.  For example, the graph of the equation $x=4$ includes the same input value of $4$ for all points on the line, but would have different output values, such as $(4,-2),(4,0),(4,1),(4,5),$ etcetera. Vertical lines are NOT functions, however, since each input is related to more than one output.

Horizontal lines have a slope of zero and is represented by the form, $y=b$, where $b$ is the $y$-intercept.  A graph of the equation $y=6$ includes the same output value of 6 for all input values on the line, such as $(-2,6),(0,6),(2,6),(6,6)$, etcetera.  Horizontal lines ARE functions because the relation (set of points) has the characteristic that each input is related to exactly one output.