How to tell if a function is linear without graphing

To see if a table of values represents a linear function, check to see if there's a constant rate of change. If there is, you're looking at a linear function! This tutorial shows you how to tell if a table of values represents a linear function.

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Using a Graph
Using an Equation
Using a Table
How do you know if a function is linear or not?
How do you know if its linear or nonlinear?
How can you tell if a function is linear or nonlinear from a table?

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A linear function creates a straight line when graphed on a coordinate plane. It is made up of terms separated by a plus or minus sign. To determine if an equation is a linear function without graphing, you will need to check to see if your function has the characteristics of a linear function. Linear functions are first-degree polynomials.

Check that the y, or independent variable, is by itself on one side of the equation. If it is not, rearrange the equation so that it is. For example, given the equation 5y + 6x = 7, move the 6x term to the other side of the equation by subtracting it from both sides. This yields 5y = 7 - 6x. Then divide both side by 5 so you have y = 7/5 - (6/5)x.

Determine whether the equation is a polynomial or not. For an equation to be a polynomial, the power of the independent or "x" variable of each term must be a whole number. The terms can be made up of constants and variables. If the equation is not a polynomial, it is not a linear equation. In the example, y = 7/5 - (6/5)x has one "x" term and its power is 1. Because 1 is a whole number, y = 7/5 - (6/5)x is a polynomial.

Determine whether the equation is a first-degree polynomial. Locate the exponent with the highest degree out of the terms. That exponent is the degree of the polynomial. If it is one, it is a linear equation. Because the highest power of "x" in y = 7/5 - (6/5)x is 1, it is a linear function.

Tips

Make sure no variable is multiplied by another variable in the function. If that's the case, it's not a linear equation.

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References

The Math Page: Polynomial Functions
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Tips

Make sure no variable is multiplied by another variable in the function. If that's the case, it's not a linear equation.

About the Author

I have a bachelor’s degree in mathematics from OSU and have written numerous articles on mathematics for eHow. I also have over 5 years experience in computer software/hardware troubleshooting. I have written many software troubleshooting documents as well as user guides for software packages such as MS Office and popular media software.

Equations are mathematical statements, often using variables, that express the equality of two algebraic expressions. Linear statements look like lines when they are graphed and have a constant slope. Nonlinear equations appear curved when graphed and do not have a constant slope. Several methods exist for determining whether an equation is linear or nonlinear, including graphing, solving an equation and making a table of values.

Using a Graph

Plot the equation as a graph if you have not been given a graph.

Determine whether the line is straight or curved.

If the line is straight, the equation is linear. If it is curved, it is a nonlinear equation.

Using an Equation

Simplify the equation as closely as possible to the form of y = mx + b.

Check to see if your equation has exponents. If it has exponents, it is nonlinear.

If your equation has no exponents, it is linear. "M" represents the slope.

Graph the equation to check your work. If the line is curved, it is nonlinear. If it is straight, it is linear.

Using a Table

Make a table of sample x values and solve for the resulting y values. Choose x values that are a constant numerical distance from each other. For example, put x values of -4, -2, 2 and 4 into the equation and solve for y for each value.

Calculate the differences between the y values.

If the differences are constant, or the same value, the equation is linear and has a constant slope. If the differences are not the same, the equation is not linear.

Tips

When simplifying equations, remember the cardinal rule: always do the same thing to both sides.

Warnings

Some slightly curving graphs may appear linear at first glance. Check a graph's linearity by finding its slope at several points. If the points have the same slope, the equation is linear. If the graph does not have a constant slope, it is not linear.

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References

Purple Math: Solving One-Step Linear Equations
Kwiz Net: Linear Equations
Purple Math: Systems of Non-Linear Equations

Tips

When simplifying equations, remember the cardinal rule: always do the same thing to both sides.

Warnings

Some slightly curving graphs may appear linear at first glance. Check a graph's linearity by finding its slope at several points. If the points have the same slope, the equation is linear. If the graph does not have a constant slope, it is not linear.

About the Author

Rebekah Richards is a professional writer with work published in the "Atlanta Journal-Constitution," "Brandeis University Law Journal" and online at tolerance.org. She graduated magna cum laude from Brandeis University with bachelor's degrees in creative writing, English/American literature and international studies. Richards earned a master's degree at Carnegie Mellon University.

How do you know if a function is linear or not?

So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value. For example, for any one-step change in x, is the change in y always going to be 3? Is it always going to be 5? If it's always going to be the same value, you're dealing with a linear function.

How do you know if its linear or nonlinear?

A linear equation forms a straight line on a graph. A nonlinear equation forms an S-curve, bell curve or another nonlinear shape on a graph. Professionals in mathematics and physics view linear equations as simple. Professionals in mathematics and physics view nonlinear equations as complicated.