Linear function

In basic mathematics, a linear function is a function whose graph is a straight line in 2-dimensions (2D, see images). An example is: "y"=2"x"–1. In higher mathematics, a linear function often refers to a linear mapping.
Basic Properties.
Formally, a linear function is a function "f"("x"):R→R such that the graph of "f" is a line. This means the domain or input of "f" is a real number R and the range or output of "f" is also a real number R. Usually we write "y"("x") or just "y" in place of "f"("x"). So the formal statement means:
There are three main forms for writing linear functions: slope-intercept, standard and parametric.
Slope-intercept form.
The slope-intercept (also called point-slope or explicit) form of a linear function is  formula_1  or  formula_2 . This form has 2 variables "x" and "у" and 2 constants "m" and "b".
Example: "y"=–2"x"+4. The slope is "m"= –2 and the "y"-intercept is "b"=4 or the point (0,4). Substituting "y"=0 and solving for "x", we get 0=–2"x"+4 or "x"=2. So "x"=2 is the root of this linear function and the point (2,0) is the "x"-intercept. Since the slope is "m" = –2, the line is decreasing. Since |–2|=2&gt;1, the decrease is "relatively" steep. For each change in "х" of 1 (to the right), the value of "у" changes -2 (goes down).
Example: "y"=–2"x"+4. Substituting "x"=0 we get "y"=4 (this is the "y"-intercept) and thus the point (0,4). Substituting "x"=1, we get "y"=2 and thus the point (1,2). Plot these points and draw the line. (Notice that the 2nd point is 1 to the right and 2 down from the 1st point. As we said in the above example, this happens because the slope is "m"= –2)
Example: "y"= –"x"+2. Suppose "y"= –1. We substitute "y"= –1 and get: –1= –'x"+2 or "x"=3. This is the only solution. We can do this for any "y"-value.
Standard form.
   formula_3.
Example: The linear function 3"x"–2"y"=1 is in standard form. The constants are "A"=3, "B"=–2 and "C"=1.
Example: The lines 3"x"–2"y"=1 and 6"x"–4"y"=2 are coincident (same line). Here the factor is: "k"=2. We multiplied the first equation by 2 to get the second equation. The unique slope-intercept form of this line is: "y"=1.5"x"–0.5 (solve either equation for y).
Vector-Parametric form.
Parametric form: formula_5    or
Vector form: formula_6  or  formula_7 .
Example: X=(–1,1)+"t"(2,3), t∈R is a line in vector form. Here: "a"1=2, "a"2=3, "x"1=–1 and "x"2=1. The line goes through the points ("x"1,"y"1)=(–1,1) and ("x"1+"a"1,"y"1+"a"2)=(1,4). The corresponding parametric form of this line is: "x"("t")= –1+2"t", "y"("t")=1+3"t". The unique slope-intercept form of this line is: "y"("x")=1.5"x"+2.5   (solve the first equation for "t" and substitute this result into the second equation).
Example: X=(–1,1,2)+"t"(2,3,–1), t∈R is a line in 3-dimensional space. The line goes through the points (–1,1,2) and (1,4,1).
Derivative of a linear function.
In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values. A linear function has a constant rate of change. This rate of change is the slope "m". So "m" is the derivative. This is often written:
Example: "y"= –2"x"+4. Here "m"= –2 and so "y"′= –2.
Linear function vs. Linear equation.
Often, the terms "linear equation" and "linear function" are confused. Both are polynomials. However, the word "linear" in "linear equation" means that all terms with variables are first degree. (The word "linear" in "linear function" means the graph is a line.) A linear equation can have 1, 2, 3, or more variables. So a linear equation is a linear function only if it has exactly 2 variables. (A linear equation in one variable is a point on the number line and a linear equation in 3 variables is a plane in 3-dimensional space.)
Notation.
Many countries and disciplines use different letters and ordering for the different forms.
In many countries, a linear functions is often written as formula_9 where "a" is the slope and "b" is the "y"-intercept.
In business and economics, a linear function is often written as formula_10 where "a" is the "y"-intercept and "b" is the slope.