Definition: A linear function is a function whose graph is a slanted¹ line in the plane.

Properties of Linear Functions

A linear function is a polynomial function of first degree with one independent variable х, i.e. $y(x)=ax+b$ To use the function or graph the line, the coefficient letters a and b must be given as actual real numbers. For example: $y(x)=2x-1$ . Here a=2 and b=-1. In the function, x is the independent variable and y is the dependent variable. The domain or set of allowed values for x of a linear function is $\Re$ (all real numbers). This means that any real number can be substituted for x. (Of course, the value of y depends on the substituted value for x.) The set of all points: (x,y(x)) is the line. Since two points determine a line, it is enough to substitute two different values for x in the linear function and determine y for each of these values (see videos below).
	Graph of the linear function: y(x)=-x+2
Because the graph of a linear function is a slanted line: A linear function has exactly one intersection point with the у-axis. This point is (0,b). A linear function has exactly one intersection point with the х-axis. This point is (-b⁄a,0). From this, we get that a linear function has exactly one zero or root. That is, there is exactly one solution to the equation ax+b=0. The zero is x=-b⁄a. There are three standard forms for linear functions. General form Slope-intercept form Vector-parametric form Quite often the term linear equation is used interchangably with linear function. While a linear function in general form is indeed a linear equation, the opposite is definitely not true.

General Form

$Ax+By=C$ where $A\neq 0$ and $B\neq 0$ .

Properties of the general form

The general form has 2 variables x and у and 3 coefficient letters A, B, and C.
To use the function or graph the line, the coefficient letters A, B and C must be given as actual real numbers: 3x-2y=1. Here A=2, B=-2 and C=1.
This form is not unique. If we multiply A, B and C by a factor k, we will have the same line.
- Example: with k=3 we have that 3x-2y=1 and 9x-6y=3 are the same line.
- Example: with k=-1⁄π we have that -3πx+2πy+π=0 and 3x-2y=1 are the same line.
This form is used mainly in geometry and in systems of two linear equations in two unknowns.
The general form of a line is a linear equation; the opposite is not necessarily true.

Example: 3x-2y=1 and 6x-4y=2 are the same linear function, i.e. their graph is the same line.

In the first equation the coefficients are: A=3, B=-2 and C=1.
In the second equation the coefficients are: A=6, B=-4 and C=2.
Notice that the second coefficients are all twice the first equations. This means the factor is k=2.
Further, solving both of these equations for y gives the same slope-intercept form of this line.
y=1,5x-0,5

Slope-Intercept Form

$y(x)=ax+b$ or $y=ax+b$ where ${a\neq 0}$ .

Properties of the Slope-Intercept Form

The slope-intercept form is also called the explicit form because it defines y(x) explicitly (directly) in terms of x.
The slope-intercept form has 2 variables x and у and 2 coefficient letters а and b.
To use the function or graph the line, the coefficient letters must be written as actual real numbers. For example: y(х)=-2х+4
The slope-intercept form is unique. That is, if we change the value of either or both a and b, we get a different line!
Every linear function can be written uniquely in slope-intercept form.
Intercepts (intersections of the line with the axes)
- The constant b is the so-called у-intercept. It is the y-value at which the line intersects the y-axis. This is because the y-axis is the line where x=0 and if we substitute x=0 into the linear function y(x)=ax+b we get: y(0)=a•0+b=b. This means that the point (0,b) is both a point on the line and a point on the y-axis. So it is the point where the line intersects the y-axis.
- The number -b⁄а is the [root] or [zero] of the function. It is the x-value at which the line intersects the x-axis. This is because the x-axis is the line where y=0 and if we substitute y=0 into the linear function and solve (backwards) for x, we get:
         0=a•x+b
         a•x+b=0
         a•x = -b
         x=-b⁄a
  This means that the point (-b⁄a,0) is both a point on the line and a point on the x-axis. So it is the point where the line intersects the x-axis.

The coefficient а is the so-called slope of the line and is a measure of the rate of change of the linear function. Since a is a number (not a variable), this rate of change is constant. For every increase in x by 1, the value of the function changes by a.

Example: $y(x)=-2x+4$ where

a = -2 and b = 4
(0,b) = (0,4) is the intersection of the line and the у-axis
(-b⁄а,0) = (-4⁄-2,0) = (2,0) is the intersection of the line and the х-axis and
а = -2 is the slope of the line. For every step to the right (х increases by 1), the value of у changes by -2 (goes down).

Vector-Parametric Form

Parametri: $\left\{{\begin{array}{*{20}{l}}{x(t)={b_{1}}+{a_{1}}t}\\{y(t)={b_{2}}+{a_{2}}t}\end{array}}\right.$ or Vector: ${X}=({b_{1}},{b_{2}})+t({a_{1}},{a_{2}})$ where ${a_{1}\neq 0}$ and ${a_{2}\neq 0}$ .

Properties of the vector-parametric form

Vector-parametric form has 1 parameter t, 2 variables x and у, and 4 coefficients а₁, а₂, b₁, and b₂.
The coefficients are not unique, but they are related.
The line passes through the points (b₁,b₂) and (b₁+a₁,b₂+a₂).
The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with t=time).
Engineers tend to use parametric notation and the letter t for the parameter; mathematicians use vector notation and the letter λ.
This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.

Example: ${X}=({-1},{1})+t({2},{3})$

We have: a₁ = 2 and a₂ = 3 and b₁ = -1 and b₂ = 1
The line passes through the points> (b₁, b₂)=(-1,1) and (b₁+a₁,b₂+a₂)=(1,4)
These are the points where t=0 and t=1 (t is not visible on the graph!).
The parametric form of this line is: $\left\{{\begin{array}{*{20}{l}}{x(t)={-1}+{2}t}\\{y(t)={1}+{3}t}\end{array}}\right.$
The slope-intercept form of this line is: y(x)=1,5x+2,5 (solve the first parametric equation for t and substitute into the second).
One general form of this line is: -3x+2y=5.

Video - How to sketch the graph of Ax+By=C

Video - How to sketch the graph of y(x)=ax+b

¹ Slanted meaning neither vertical nor horizontal.

References