Topic 1.6 Notes – Polynomial Functions and End Behavior

What End Behavior Is

For a polynomial $p(x)$, end behavior asks two separate questions:

• What happens as $x \to \infty$ → the right end
• What happens as $x \to -\infty$ → the left end

The output can only do one of two things at each end:

$$\lim_{x\to\infty}p(x)=\infty \qquad \lim_{x\to\infty}p(x)=-\infty$$

$$\lim_{x\to-\infty}p(x)=\infty \qquad \lim_{x\to-\infty}p(x)=-\infty$$

A graph rises if outputs go to $\infty$. It falls if outputs go to $-\infty$.

That’s all long-run behavior. The middle of the graph can wiggle, cross the axis, and turn around a lot. None of that changes the ends.

These four sketches show the standard end-behavior patterns you’ll use over and over.

Common polynomial end-behavior patterns

A quick reminder from earlier polynomial work: when $|x|$ gets very large, the highest-power term matters most.

The Leading Term Controls the Ends

A polynomial in standard form looks like

$$p(x)=a_nx^n+\cdots+a_0$$

Here:

• leading term is $a_nx^n$
• leading coefficient is $a_n$
• degree is $n$

Lower-degree terms matter in the middle, but not at the ends, because $x^n$ grows much faster than $x^{n-1}, x^{n-2}$, and constants when $|x|$ is large.

The only two features you need are:

• parity of degree
  • even degree → both ends go the same way
  • odd degree → ends go opposite ways
• sign of leading coefficient
  • positive → right end rises
  • negative → right end falls

The four cases

Fast shortcut to memorize:

• even = same on both ends
• odd = opposite ends
• the sign tells the right end

Finding End Behavior from an Equation

When the polynomial is already in standard form, the process is quick:

1. Find the term with the greatest exponent.
2. Decide whether that exponent is even or odd.
3. Check whether its coefficient is positive or negative.
4. Use the four-case chart.

For $q(x)=-5x^7+3x^4-8x^2+12$, the leading term is $-5x^7$.

• degree $7$ → odd
• coefficient $-5$ → negative

So the graph rises left and falls right:

$$\lim_{x\to-\infty}q(x)=\infty \qquad \lim_{x\to\infty}q(x)=-\infty$$

If the polynomial is factored, build the leading term from the factors. For

$$r(x)=-3(x-2)^2(x+1)^3$$

the leading parts are $x^2$ and $x^3$, so the overall leading term is

$$-3(x^2)(x^3)=-3x^5$$

That gives odd degree, negative coefficient, so again:

• rises to the left
• falls to the right

On a quiz, write the conclusion either in limit notation or in graph words.

Reading End Behavior from a Graph or Table

From a graph, look only at the far left and far right. Ignore turning points in the middle.

• both ends up → even degree, positive leading coefficient
• both ends down → even degree, negative leading coefficient
• left down, right up → odd degree, positive leading coefficient
• left up, right down → odd degree, negative leading coefficient

You can infer parity and sign, but not exact degree. A quadratic and a quartic can both rise on both ends.

From a table, very large positive and negative inputs can suggest end behavior, but a finite table is only evidence unless you know the actual polynomial rule.

Also be careful with graphing windows. A calculator can hide the true far-end behavior.

Common Mistakes and Edge Cases

• Using a lower-degree term instead of the leading term.
• Mixing up left end with right end.
• Confusing $x \to -\infty$ with the graph going down. Those are different. $x$ tells input direction; the limit tells output direction.
• Letting middle behavior decide the ends.
• Claiming the exact degree from end behavior alone.
• Forgetting constants:
  • if $p(x)=c$, then both limits equal $c$
  • if $p(x)=0$, the output is always $0$