# Linear function

Before we define linear function let’s first define few terms. We all know what a number line is, but now we’ll have to go a step further and define a number plane.

A number plane has two axis, horizontal and vertical. Vertical is called the $y$-axis, and horizontal $x$-axis. It is also known as Cartesian plane.

Their intersection will be marked with point $O$, on the $x$-axis, numbers on the left will be negative and on the right positive, and on $y$-axis, numbers below $0$ will be negative and above positive.

A coordinate is an ordered pair of numbers which tells us where specific point is located. First number indicates where the point is located considering $x$- axis, and second considering $y$- axis. The plain is divided into quadrants.

Linear function is a function given by a rule $f(x) = a \cdot x$, where $a$ is from $a$ set of real numbers. In our examples $f (x)$, placed on the bottom of this lessons, will be replaced with $y$. In this rule, $x$ is the changeable variable. That means that you can take any numbers in the place of $x$ and get yourself an ordered pair of numbers. Every function is represented by a graph. A graph is a set of all ordered pairs that satisfy rule of a function. For linear functions that graph is a straight line that goes trough the $(0, 0)$ coordinate. There are two parts of every linear function, the dependent variable, or $f(x)$ in this case, and independant variable, $x$. This only means that you take x arbitrarily, and your $f(x)$ depends on your choice.

Example: Draw a graph of linear function: $f(x) = 2x$
First you have to make your dependant/independant variable table.

This is how it will usually looks like. You only need two points to make a line, but for precise in drawings, this time we’ll take three points.

Now, how to get $f(x)$. You have your function rule. In that rule you “put you $x$ to get $f(x)$”.
$f(x) = 2x$ => $f(-1)= 2(-1) = -2$ => $f(0) = 2 * 0 = 0$ $f(1) = 2 * 1 = 2$

Now you have your ordered pairs : $(-1,-2), (0,0), (1,2)$ .All you have to do to make a graph is put them in the plane, and connect them with a straight line. So how do you put them in a plane?

Now we have linear function graph which is a line.

The number infront of variable $x$ indicates the slope of your line.

## Slope

The slope of a line is a number that describes steepness and direction of the line.

If we have two points:

$A=(x_1,y_1)$

$B=(x_2,y_2)$

A slope ($a$) is calculated by the formula:

$a=\frac{y_2-y_1}{x_2-x_1}$

If the slope is equal to number $0$, then the line will be paralel with $x$ – axis. $f(x) = b$.
If variable $x$ is a constant $x =c$, that will represent a line paralel to $y$-axis.

## Worksheets

Linear function

Linear functions - Point-slope form (161.7 KiB, 788 hits)

Linear function - Slope-intercept form (208.7 KiB, 769 hits)

Linear functions - Standard form (972.7 KiB, 698 hits)

Graphing linear functions (2.0 MiB, 947 hits)

Slope

Determine slope in slope-intercept form (160.4 KiB, 666 hits)

Determine slope from given graph (2.1 MiB, 661 hits)

Find the integer of unknown coordinate (273.6 KiB, 696 hits)

Find the fraction of unknown coordinate (418.5 KiB, 776 hits)

Linear inequalities

Graph of linear inequality (2.8 MiB, 761 hits)