Overview
In order for functions to be linear, they must follow two criteria. First, each domain value must be paired with one range value. If a domain value is paired with more than one range value, the relationship is not a function. Second, the paired domain and range values must fall on a line to be a linear function.
Identifying Functions
In order for a relationship to be a function, each domain value (x) must be paired with a range value (y). Suppose that a particular set has ordered pairs of {(4, -4) (3,-2) (2, 0) (1, 2)}. The domain values are (4, 3, 2, 1) paired with range values of (-4, -2, 0, 2). It is a function. Another set has values of { (3, 0) (3, 1) (3, 2) (3, 3)}. It is not a function, because the domain value 3 is paired with different range values (0, 1, 2, 3).
Linear Functions
For functions to be linear, the change in the domain values must be constant and the change in the range values must also be constant. The set with ordered pairs {(4, -4) (3, -2) (2, 0) (1,2)} has a constant change in the domain values, as 4-1 is 3, 3-1 is 2, and 2-1 is 1. It also has a constant change in the range values as -4 +2 is -2, -2 +2 is 0, and 0 +2 is 2. Not all functions are linear, however. A set such as {(-2, 6) (-1, 3) (0, 2) (1, 3) (2, 6)} is a function, with a constant change in the domain values, as -2 +1 is -1, -1 +1 is 0, 0 +1 is 1, and 1 +1 is 2. The change in the range values is not constant, as 6-3 is 3, 3-2 is 1, 2 +1 is 3, and 3 +3 is 6.
Standard Linear Equation
Linear equations can be written in a standard form ax + by = c, where a, b, and c are all real numbers, and a and b are not both 0. Notice that this means that either a or b can be 0. For example, an equation such 3x +5y = 16 is already in standard form, as a equals 3, b equals 5, and c equals 16. Similarly, an equation such as y = 5x – 3 can be put in standard form as -5x +y =-3.
Graphing Linear Functions
If the equation for a function can be written in standard form, it can be graphed as a linear function. Suppose the equation is 2x +3y = 20. It is already in standard form, as a =2, b =3, and c =20. If x equals 1, then y equals 6, if x equals 4, then y equals 4, and if x equals 7, then y equals 2.
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