Linear vs. Nonlinear Functions

Functions can be classified in two different categories: linear or nonlinear. A simple way to know differentiate between the two is to look at the output values when plugging in a number for an unknown variable. If the output values have a difference that is constant, then the function will be classified as a linear equation; however, if the outputs do not have a constant difference then it will be classified as a nonlinear equation.

Another easy way to determine which of these functions you are dealing with is to graph it. By graphing these functions, you can tell if the line is straight or not. When graphed, a linear equation will have a straight line that has a constant slope. In contrast to this, a nonlinear equation will have a graph that does not have a straight line and, depending on the function, can have many different appearances including a U-shape or an S-shape.

Look at this example:

Equation: 2x+6=y As you can see in this example, the differences between the outputs (y) are constant with each increasing number plugged into x. Also, when you graph this equation you will see a straight line that shows that as x increases by 1, y increases by 2; therefore, this is a linear equation.

Compare to this example:

Equation: x2 In this equation, the output values (y) are not constant because they are increasing at different rates with each increasing value of x. When graphing this function, you will get a U-shaped graph that is very obviously not a straight line. As you can see here, x and y do not increase consistently and therefore this is a nonlinear equation.

These functions may seem hard to differentiate at first, but by breaking it down and looking at the results of plugging in numbers for x and y, it will be easy to tell what kind of equation you are working with. Simply look for a constant increase in y after plugging in values for x, or graph the equation and look for a straight line. A linear equation will have constantly increasing y values and a straight line, while a nonlinear equation will have outputs increasing at a non-constant rate and a curved graph. After awhile, determining these functions will become easy and you will be able to tell which function you have simply by looking at the equation itself. 