End behavior is a concept in mathematics that helps us understand what a polynomial function does when the input values increase or decrease without bound. In other words, it describes how the function behaves as we move towards positive or negative infinity on the x-axis. When input values increase or decrease without bound, the leading term (the term with the highest degree) dominates the values of all lower-degree terms in the polynomial function. This dominance of the leading term is crucial because it ultimately determines the end behavior of the function.

The degree and sign of the leading term play a significant role in defining the end behavior of a polynomial function. To represent the end behavior as input values increase without bound, we use the notation lim_{x\to \infty} p(x) = \infty or lim_{x\to \infty} p(x) = -\infty. This notation helps us understand whether the function approaches positive infinity or negative infinity as the input values approach positive infinity.

Similarly, when we want to describe the end behavior as input values decrease without bound, we use the notation lim_{x\to -\infty} p(x) = \infty or lim_{x\to -\infty} p(x) = -\infty. This notation indicates whether the function approaches positive infinity or negative infinity as the input values approach negative infinity.

By analyzing the degree and sign of the leading term and understanding these notations, we can effectively describe and interpret the end behavior of polynomial functions.

Degree	Leading Coefficient	End Behavior	Graph of the function
\text{even}	\text{positive}	\lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty	\text{rises to the left and} \\ \text{to the right}
\text{even}	\text{negative}	\lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty	\text{falls to the left and} \\ \text{to the right}
\text{odd}	\text{positive}	\lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty	\text{falls to the left and} \\ \text{rises to the right}
\text{odd}	\text{negative}	\lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty	\text{rises to the left and} \\ \text{falls to the right}

Examples

Example 1

Describe the end behavior for each of the following functions.

p\left(x\right) = -2x^4 + 3x^3 - 5x^2 + x + 1.

Worked Solution

Create a strategy

Analyze the degree and sign of the leading term to describe the end behavior.

Apply the idea

The leading term of p\left(x\right) is -2x^4. The degree is 4, which is even, and the leading coefficient is -2, which is negative.

For even-degree functions, both ends of the function behave similarly. Since the leading coefficient is negative, both ends approach negative infinity.

In other words, as x approaches positive infinity, p(x) approaches negative infinity, and as x approaches negative infinity, p(x) approaches negative infinity.

This can be written in limit notation as:

\lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to \infty} f(x) \to - \infty

f\left(t\right) = t^3 - 2t^2 + t - 1

Worked Solution

Create a strategy

Analyze the degree and sign of the leading term to describe the end behavior.

Apply the idea

Leading term: t^3

Degree: 3

Sign: +

End behavior:

\lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to \infty} f(x) \to \infty

h\left(k\right)=2k+4-\dfrac{1}{2}k^5

Worked Solution

Create a strategy

Analyze the degree and sign of the leading term to describe the end behavior.

Apply the idea

Leading term: -\dfrac{1}{2}k^5

Degree: 5

Sign: -

End behavior:

\lim_{x \to -\infty} f(x) \to \infty \\ \lim_{ x \to \infty} f(x) \to - \infty

w\left(x\right)=x^3+2x^6

Worked Solution

Create a strategy

Analyze the degree and sign of the leading term to describe the end behavior.

Apply the idea

Leading term: 2x^6

Degree: 6

Sign: +

End behavior:

\lim_{x \to -\infty} f(x) \to \infty \\ \lim_{ x \to \infty} f(x) \to \infty

Example 2

Consider the end behavior for a function p\left(x\right):

\lim_{x \to \infty} p(x) \to \infty \\ \lim_{ x \to -\infty} p(x) \to - \infty

Describe the end behavior of p\left(x\right) in words.

Worked Solution

Apply the idea

The function rises to the right and falls to the left.

State whether p\left(x\right) has an even or odd degree. Explain.

Worked Solution

Apply the idea

p\left(x\right) has an odd degree because the ends have different behavior.

State the sign of the leading coefficient of p\left(x\right). Explain.

Worked Solution

Apply the idea

The leading coefficient is positive because the function rises to the right.

Idea summary

Degree	Leading Coefficient	End Behavior	Graph of the function
\text{even}	\text{positive}	\lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty	\text{rises to the left and} \\ \text{to the right}
\text{even}	\text{negative}	\lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty	\text{falls to the left and} \\ \text{to the right}
\text{odd}	\text{positive}	\lim_{x \to -\infty} f(x) \to - \infty \\ \lim_{ x \to +\infty} f(x) \to + \infty	\text{falls to the left and} \\ \text{rises to the right}
\text{odd}	\text{negative}	\lim_{x \to -\infty} f(x) \to + \infty \\ \lim_{ x \to +\infty} f(x) \to - \infty	\text{rises to the left and} \\ \text{falls to the right}

1.6 Polynomial functions and end behavior

Introduction

Ideas

End behavior of polynomials

Examples

Example 1

Example 2

Outcomes

1.6.A

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