Math Practice

End Behavior Practice Problems

End behavior comes from two facts and nothing else: whether the degree is even or odd, and whether the leading coefficient is positive or negative. These twelve problems drill that pair, including the cases where you have to find the leading term yourself from a factored or rearranged expression.

Questions

1.
What is the degree of ?
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The degree is the highest power present, which is 4. The other terms have no effect on end behavior at all — only the leading term matters once gets large.
2.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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Odd degree with a positive leading coefficient: the right tail rises. Cubing a large positive number gives a larger positive number, so .
3.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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The squaring makes the result positive, but the flips it negative. So and the tail falls. With an even degree both tails do the same thing, so the left one falls too.
4.
has both tails going the same way. Which way? Answer "up" or "down".
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Degree 4 is even, so the tails match, and the leading coefficient 5 is positive, so they both go up. The term shapes the middle of the graph but is overwhelmed at the extremes.
5.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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Take large and negative. Then is large and negative, and the minus sign in front makes large and positive. So the left tail rises. Odd degree with a negative leading coefficient always gives up on the left, down on the right.
6.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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Degree 5 is odd and the leading coefficient 4 is positive, so the tails disagree: down on the left, up on the right. A large negative raised to the fifth power stays negative, and multiplying by 4 keeps it there.
7.
What is the greatest number of turning points a degree-6 polynomial can have?
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At most , so . It can have fewer — has exactly one — but never more. The count is a useful check on a sketch.
8.
For , which way do both tails go? Answer "up" or "down".
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You do not need to expand it. Multiply only the leading terms of each factor: . Degree 4 is even and the coefficient is , so both tails go up.
9.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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Add the exponents for the degree: , which is odd. The leading coefficient is the out front times the from each bracket, so it is negative. Odd degree with a negative lead falls on the right.
10.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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Watch the first bracket: its leading term is , not . Multiplying leading terms gives . Odd degree, negative coefficient — so the left tail rises. A factor written with the constant first is the classic trap here.
11.
For , as , does the graph rise or fall? Answer "rise" or "fall".
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The degree is , which is even, and the leading coefficient is . Even degree with a positive lead sends both tails up, so the left one rises.
12.
A polynomial has both tails going down. Is its degree even or odd? Answer "even" or "odd".
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Matching tails means an even degree — an odd-degree polynomial always has its two tails going opposite ways. Both tails going down additionally tells you the leading coefficient is negative.

Written by

Zohaib

MPhil in Mathematics, Founder of Mathovia

Published

11 September 2026

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End Behavior Practice Problems

The end behavior lesson reduces the whole topic to a two-by-two table. These twelve problems drill it, with the harder ones requiring you to work out the leading term yourself before the table applies.

The four cases

Degree	Leading coefficient	Left tail	Right tail
even	positive	up	up
even	negative	down	down
odd	positive	down	up
odd	negative	up	down

Finding the leading term without expanding

For a factored polynomial, multiply only the leading term of each factor:

Degree 5, coefficient . That is everything the table needs, and it took one line instead of expanding a quintic.

Watch for a factor written backwards. has leading term , not , and it flips the sign of the whole leading coefficient. Rewriting it as before you start makes the minus sign impossible to miss.

Turning points as a check

A degree-6 polynomial can turn at most five times. This does not affect the tails, but it catches a sketch that has the right end behavior and an impossible middle.

Frequently Asked Questions

What determines a polynomial's end behavior?+

Only two things: whether the degree is even or odd, and the sign of the leading coefficient. Every other term is irrelevant, because as x grows large the leading term dominates everything else by an arbitrarily wide margin.

What are the four possible cases?+

Even degree with a positive lead sends both tails up. Even with a negative lead sends both down. Odd with a positive lead goes down on the left and up on the right. Odd with a negative lead does the reverse.

How do I find the leading term of a factored polynomial?+

Multiply the leading terms of each factor together and ignore everything else. For (x − 1)(x + 2)(x − 3) that is x · x · x = x³. You never need to expand the whole product just to read end behavior.

What if a factor is written as (2 − x) instead of (x − 2)?+

Its leading term is −x, so it contributes a factor of −1 to the leading coefficient. This is the most common way an end-behavior answer comes out backwards, and it is worth rewriting such a factor as −(x − 2) before you start.

Do turning points affect end behavior?+

No — they are a separate feature of the middle of the graph. But the turning-point limit of n − 1 is a useful cross-check, because a sketch with too many wiggles is wrong regardless of whether the tails are right.

Why does the constant term never matter?+

Because it does not grow. At x = 1000 a constant of 7 is negligible against a leading term in the billions. End behavior is a statement about limits, and only the fastest-growing term survives it.

↗↘End Behavior Practice Problems

End Behavior Practice Problems

The four cases

Finding the leading term without expanding

Turning points as a check

Frequently Asked Questions

More practice problems

End Behavior Practice Problems