How to Find End Behavior of a Function Without Graphing

You can use the leading coefficient test to figure out end behavior of the graph of a polynomial function. This isn't some complicated theorem. There's no factoring or x-intercepts. Here are two steps you need to know when graphing polynomials for their left and right end behavior.

Defining the Terms

Let's review some common precalculus terms you'll need for the leading coefficient test:

A polynomial is a fancy way of saying "many terms."

A polynomial function is a function (a statement that describes an output for any given input) that is composed of many terms.

2x3+8-4 is a polynomial.

y=2x3+8-4 is a polynomial function.

A leading term in a polynomial function f is the term that contains the biggest exponent.

A coefficient is the number in front of the variable.

End behavior is another way of saying whether the graph ascends or descends in either direction.

The leading coefficient test is a quick and easy way to discover the end behavior of the graph of a polynomial function by looking at the term with the biggest exponent.

Step 1: The Coefficient of the Leading Term Determines Behavior to the Right

The behavior of the graph is highly dependent on the leading term because the term with the highest exponent will be the most influential term. Let's step back and explain these terms.

2x3 is the leading term of the function y=2x3+8-4.

2 is the coefficient of the leading term.

When graphing a function, the leading coefficient test is a quick way to see whether the graph rises or descends for either really large positive numbers (end behavior of the graph to the right) or really large negative numbers (end behavior of the graph to the left).

Let's start with the right side of the graph, where only positive numbers are in the place of x.

If the leading coefficient is positive, bigger inputs only make the leading term more and more positive. The graph will rise to the right.

Leading coefficient test: Desmos leading coefficient is negative

If the leading coefficient is negative, bigger inputs only make the leading term more and more negative. The graph will descend to the right.

Leading coefficient test: 2x3 leading term

Leading coefficient test: Desmos leading coefficient is negative

Leading coefficient test: 3x5 leading term

Step 2: The Degree of the Exponent Determines Behavior to the Left

The variable with the exponent is x3.

When you replace x with positive numbers, the variable with the exponent will always be positive. So you only need to look at the coefficient to determine right-hand behavior.

When you replace x with negative numbers, the variable with the exponent can be either positive or negative depending on the degree of the exponent.

A negative number multiplied by itself an odd number of times will remain negative. A negative number multiplied by itself an even number of times will become positive.

The leading coefficient test tells us that the graph rises or falls depending on whether the leading terms are positive or negative, so for left-hand behavior (negative numbers), you will need to look at both the coefficient and the degree of the component together.

Let's look at the following examples of when x is negative:

Leading coefficient test: 2x3would be a positive coefficient

Leading coefficient test: 2x4 would be a positive coefficient

Leading coefficient test: -5x3 would be a negative coefficient

Leading coefficient test: -5x2 would be a negative coefficient

A trick to determine end graphing behavior to the left is to remember that "Odd" = "Opposite." If the degree is odd, the end behavior of the graph for the left will be the opposite of the right-hand behavior. If the degree is even, the variable with the exponent will be positive and, thus, the left-hand behavior will be the same as the right.

Using the Leading Coefficient Test

When graphing a polynomial function, look at the coefficient of the leading term to tell you whether the graph rises or falls to the right.

Look at the exponent of the leading term to compare whether the left side of the graph is the opposite (odd) or the same (even) as the right side.

When in doubt, split the leading term into the coefficient and the variable with the exponent and see what happens when you substitute either a negative number (left-hand behavior) or a positive number (right-hand behavior) for x.

Finally, here are some complete examples illustrating the leading coefficient test:

Desmos graphing calculator rising to the right and descends to the left