﻿ End Behavior of Polynomials

# End Behavior of Polynomials

## End Behavior of General Functions

Roughly, the ‘end behavior’ of a function refers to what happens to its graph as you travel ...

• ... farther and farther to the right; this is called right-hand end behavior
• ... farther and farther to the left; this is called left-hand end behavior

‘Farther and farther to the right’ means you let $\,x\,$ go to infinity, which is written as ‘$\,x\rightarrow\infty\,$’.

‘Farther and farther to the left’ means you let $\,x\,$ go to negative infinity, which is written as ‘$\,x\rightarrow-\infty\,$’.

End behavior is studied to understand what happens to the outputs from a function (the $y$-values of points on the graph) as you move farther and farther away from the origin.

For example, you might want to know:

• Are the outputs getting bigger and bigger, without bound?
• Are the outputs getting closer to zero?
• Are the outputs approaching some nonzero number?

In general, functions can have a variety of end behaviors, as shown in the table below. Only right-hand end behaviors are shown here.

Increasing without bound as you move to the right:

as $x\rightarrow\infty\,,$ $\,y\rightarrow\infty$

‘as $\,x\,$ goes to infinity, $\,y\,$ goes to infinity’  as in $\,y = x^2\,$  as in $\,y = \sqrt{x}\,$  as in $\,y = x\,$
Decreasing without bound as you move to the right:

as $x\rightarrow\infty\,,$ $\,y\rightarrow-\infty$

‘as $\,x\,$ goes to infinity, $\,y\,$ goes to negative infinity’  as in $\,y = -x^2\,$  as in $\,y = -\sqrt{x}\,$  as in $\,y = -x\,$
Approaching a particular real number (like $\,3\,$) as you move to the right:

as $x\rightarrow\infty\,,$ $\,y\rightarrow 3$

‘as $\,x\,$ goes to infinity, $\,y\,$ approaches $\,3\,$’ as in $\displaystyle\,y = 3 + \frac 1x$ as in $\displaystyle\,y = 3 - \frac 1x\,$ as in $\displaystyle\,y = 3 + \frac{\sin x}{x}\,$

## End Behavior of Polynomials

Polynomials can't exhibit the full variety of end behavior illustrated above. As $\,x\,$ gets large, $\,y\,$ can only do two things: go to infinity, or go to negative infinity. The best news is that we only need one term of the polynomial to figure out what the end behavior is!

KEY IDEA:
For large values of $\,x\,,$ a polynomial ‘behaves like’ its highest power term.

For example, suppose you're investigating the end behavior of: $$P(x) = x^3 - 3x^2 + 5x - 10$$

Notice that:

• the degree of $\,P\,$ is $\,3\,$
• the leading coefficient is $\,1\,$
• the highest power term is $\,x^3\,$

To investigate the end behavior of $\,P\,,$ you can instead work with the much simpler polynomial $\,Q(x) = x^3\,.$

Why? A renaming of $\,P(x)\,$ does the trick, where we factor out the highest power of $\,x\,$:

\begin{align} P(x) &= \cssId{s61}{x^3 - 3x^2 + 5x - 10}\cr\cr &= \cssId{s62}{x^3\left(1 - \frac 3x + \frac 5{x^2} - \frac{10}{x^3}\right)} \end{align}

From this new name, it is clear that as $\,x\,$ gets big, all the terms go to zero except the highest power term. Roughly, the highest power term ‘washes out’ all the other terms when $\,x\,$ is big.

Think about this. Even when $\,x\,$ is as small as $\,x = 50\,,$ let's investigate each of the terms in $\,P(x)\,$:

• $x^3 = 50^3 = 125000$
• $-3x^2 = -3\cdot 50^2 = -7500$
• $5x = 5\cdot 50 = 250$
• $-10$
• $P(50) = 125000 {-} 7500 + 250 - 10$

How significant are those other terms compared to the first? Not! The highest power term determines the behavior of the output when the input is large.

KEY IDEA:
The leading coefficient and degree of a polynomial determine its end behavior.

All possible end behaviors of polynomials are summarized in the following table.

### End Behaviors of Polynomials

Consider a polynomial with degree $\,n\,$ and leading coefficient $\,a\,.$

Thus, the highest power term is $\,ax^n\,.$

Degree ($\,n\,$): $n$ even: $\,n = 2,\, 4,\, 6,\, \ldots$
Leading coefficient ($\,a\,$): $a \gt 0$
Example: as in $\,y = 3x^2\,$:  $a = 3\,,$ $n = 2$
End behavior:
 right-hand: as $\,x\rightarrow\infty\,,$ $y\rightarrow\infty$ left-hand: as $\,x\rightarrow-\infty\,,$ $y\rightarrow\infty$
Leading coefficient ($\,a\,$): $a \lt 0$
Example: as in $\,y = -3x^2\,$:  $a = -3\,,$ $n = 2$
End behavior:
 right-hand: as $\,x\rightarrow\infty\,,$ $y\rightarrow -\infty$ left-hand: as $\,x\rightarrow-\infty\,,$ $y\rightarrow -\infty$

Degree ($\,n\,$): $n$ odd: $\,n = 1,\, 3,\, 5,\, \ldots$
Leading coefficient ($\,a\,$): $a \gt 0$
Example: as in $\,y = 2x^3\,$:  $a = 2\,,$ $n = 3$
End behavior:
 right-hand: as $\,x\rightarrow\infty\,,$ $y\rightarrow\infty$ left-hand: as $\,x\rightarrow-\infty\,,$ $y\rightarrow -\infty$
Leading coefficient ($\,a\,$): $a \lt 0$
Example: as in $\,y = -2x^3\,$:  $a = -2\,,$ $n = 3$
End behavior:
 right-hand: as $\,x\rightarrow\infty\,,$ $y\rightarrow -\infty$ left-hand: as $\,x\rightarrow-\infty\,,$ $y\rightarrow \infty$

## Example

Let $\,P(x) = 2x^3 - 5x^7 + 6x - 5\,.$ Determine the end behavior of $\,P\,.$

Solution: The highest order term is $\,-5x^7\,.$ For large values of $\,x\,,$ $\,P(x)\approx -5x^7\,.$

Think of substituting a large positive number into $\,-5x^7\,.$ The sign of the output is: $$\cssId{s124}{(-)(+)^7 = (-)}$$ The output is a large negative number, so:

as $\,x\rightarrow\infty\,,$ $\,P(x)\rightarrow -\infty$

Think of substituting a large negative number into $\,-5x^7\,.$ The sign of the output is: $$\cssId{s129}{(-)(-)^7 = (-)(-) = (+)}$$ The output is a large positive number, so:

as $\,x\rightarrow-\infty\,,$ $\,P(x)\rightarrow \infty$

## Making End Behavior Precise

Mathematical sentences like ‘as $\,x\rightarrow\infty\,,$ $\,y\rightarrow\infty\,$’ can be made precise in a Calculus course. For those of you who are curious right now:

$\text{as } x\rightarrow\infty,\ y\rightarrow\infty$

is equivalent to

for every $\,N \gt 0\,$
there exists $\,M \gt 0\,$
such that whenever $\,x \gt M\,,$
$y \gt N$