Multiplicity and End Behavior

Multiplicity, End Behavior and Graphing Polynomials

Factored form of a Polynomial

By taking a closer look at the factors we can determine the multiplicity of each factor and its corresponding root

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​Looking at the factors

determines the behavior of the graph at the zeros.

​Multiplicity

An even (2, 4, 6, 8, ...) multiplicity indicates that a zero is a turning point in the graph.


An odd (1, 3, 5, 7, ...) multiplicity indicates that the the graph will cross over the x-axis at the zero.

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End Behavior

End behavior indicates which direction the arrows at each end are pointing. There are two ends that must be described individually.

Limits are a helpful way to mathematically describe behavior of the output of a graph when approaching any input value.

When thinking about the ends of a graph, we can use limits at infinity. These describe the expected output when x is getting closer to positive or negative infinity.

the f(x) in the notation represents a function. It can be replaced with the equivalent expression or equation.

Limit notation

Cause: Indicating which direction x is going:

For limits at infinity, the x value will approach either positive or negative infinity.

x→∞

Remember that x is the value we can manipulate, y is only the result.

Limit notation

Effect: Stating the y value as a result:

For limits at infinity, the y value can approach any value; however, for polynomials, it will approach positive or negative infinity

Remember that x is the value we can manipulate, y is only the result.

∞

x→∞

End Behavior of polynomials