Поведінка цілої раціональної функції на нескоінченності

(Video) End Behavior of Polynomial Functions

by Dan Ozimek

In this video by Dan Ozimek, the concept of "end behavior" when referring to polynomials is described. This is important for understanding what the shape of a graph should look like.

(Video) End Behavior of Polynomial Functions

by NotesCollegeAlgebra

In this video by NotesCollegeAlgebra the end behavior of specific polynomial functions are shown.

Polynomial End Behavior

The end behavior of a polynomial is exactly what it sounds like: we want to know what happens to a polynomial function toward the "ends" of the $x$-axis. That means we want to describe the behavior of a polynomial as it reaches positive and negative infinity.

As we approach positive infinity on the $x$-axis, we the polynomial function can either increase or decrease. The same concept applies to a polynomial approaching negative infinity on the $x$-axis: it can either increase or decrease.

We will look at two ways to find the end behavior: graphically and algebraically.

####Graphically

Let's look at the graph of $y=x^3 + 1$.

Let's first look at the polynomial approaching positive infinity. As the values on the $x$ axis get bigger, the values in the polynomial also get bigger. As $x$ approaches positive infinity, $f(x)$ approaches positive infinity. Mathematically, we write:

$$x \rightarrow +\infty, f(x) \rightarrow +\infty$$

We can do the same analysis for the other side of the $y$-axis. As the values on the $x$ axis get smaller, the values in the polynomial also get smaller. As $x$ approaches negative infinity, $f(x)$ approaches negative infinity. Mathematially, we write:

$$x \rightarrow -\infty, f(x) \rightarrow -\infty$$

####Algebraically

Sometimes, we don't have the graph of the polynomial available to us, but we still want to know the end behavior of a polynomial. We can use the degree of the monomial ($n$) and the leading coefficient ($a$) to help us.

$$f(x) = ax^n$$

If $n$ is even: the behavior at both ends of the polynomial are the same.

• If $a$ is positive: both ends will approach positive infinity.
• If $a$ is negative: both ends will approach negative infinity.

If $n$ is odd: the behavior at both ends of the polynomial are opposites.

• If $a$ is positive: $x \rightarrow +\infty, f(x) \rightarrow +\infty$ and $x \rightarrow -\infty, f(x) \rightarrow -\infty$
• If $a$ is negative: $x \rightarrow +\infty, f(x) \rightarrow -\infty$ and $x \rightarrow -\infty, f(x) \rightarrow +\infty$

####Connecting the Dots

We can see that in the above example, we looked at a continuous function that approaches infinity. It has one turning point, with no global minimum or global maximum. It intersects the $x$-axis once.

Polynomial End-Behavior

In this next lesson on polynomials, you will learn about polynomial end-behavior, which is the behavior of the ends of a function. Think about where the function goes as it approaches infinity!

Memorize these rules to understand polynomial end-behavior!

If $n$ is even: the behavior at both ends of the polynomial are the same.

• If $a$ is positive: both ends will approach positive infinity.
• If $a$ is negative: both ends will approach negative infinity.

If $n$ is odd: the behavior at both ends of the polynomial are opposites.

• If $a$ is positive: $x \rightarrow +\infty, f(x) \rightarrow +\infty$ and $x \rightarrow -\infty, f(x) \rightarrow -\infty$
• If $a$ is negative: $x \rightarrow +\infty, f(x) \rightarrow -\infty$ and $x \rightarrow -\infty, f(x) \rightarrow +\infty$