f(x) = x^4 + 5x^3 + 7x^2

f(x) to infinity as x to -infinity, f(x) to infinity as x to infinity

f(x) = -2x^3 - x^2 - 4

f(x) to infinity as x to -infinity, f(x) to -infinity as x to infinity

f(x) to -infinity as x to -infinity, f(x) to infinity as x to infinity

f(x) = -5x^4 + 9x - 6

f(x) to -infinity as x to -infinity, f(x) to -infinity as x to infinity

f(x) = x^3 + 7x^2 - 9x

f(x) to -infinity as x to -infinity, f(x) to infinity as x to infinity

f(x) = -x^3 + 4x^2 - 8x

f(x) to infinity as x to -infinity, f(x) to -infinity as x to infinity

f(x) = 5x^4 + 6x^3 - 8x - 4

f(x) to infinity as x to -infinity, f(x) to infinity as x to infinity

f(x) to -infinity as x to -infinity, f(x) to -infinity as x to infinity

f(x) to -infinity as x to -infinity, f(x) to infinity as x to infinity

f(x) = -x^3 + 7x^2 + 9x + 2

f(x) to infinity as x to -infinity, f(x) to -infinity as x to infinity

f(x) to -infinity as x to -infinity, f(x) to infinity as x to infinity

Worksheets · Precalculus

End behavior worksheet for Precalculus

Only two things matter at the ends: the degree and the sign of the leading coefficient. Everything in the middle of the polynomial is too small to matter once x is large enough.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name End Behavior of Polynomial Functions Date Period Describe the end behavior of each function, or answer the question asked. Describe the end behavior $f (x) = x^{4} + 5 x^{3} + 7 x^{2}$ $f (x) = - 2 x^{3} - x^{2} - 4$ $f (x) = 2 x^{5} - 2 x^{4}$ $f (x) = - 5 x^{4} + 9 x - 6$ $f (x) = x^{3} + 7 x^{2} - 9 x$ $f (x) = - x^{3} + 4 x^{2} - 8 x$ $f (x) = 5 x^{4} + 6 x^{3} - 8 x - 4$ $f (x) = - 2 x^{4} - 4 x^{2}$ $f (x) = 5 x^{3} + 2 x - 1$ $f (x) = - x^{3} + 7 x^{2} + 9 x + 2$ Relate turning points and degree A polynomial's graph has 1 turning point. What is the least possible degree? A polynomial's graph has 5 turning points. What is the least possible degree? At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{5} + 6 x^{4} + x^{2} + 8 x - 2$ have? At most how many turning points can a polynomial of the same degree as $f (x) = 4 x^{6} + 9 x^{5}$ have? At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{2} + 4$ have? At most how many turning points can a polynomial of the same degree as $f (x) = - 4 x^{4} - 6 x^{2} + 2$ have? A polynomial's graph has 6 turning points. What is the least possible degree? At most how many turning points can a polynomial of the same degree as $f (x) = - 5 x^{7} + 3 x^{6} - 8 x^{5} + 6 x^{4} - 9 x^{2} - 3 x$ have? At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{3} - 7 x$ have? At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{2} - 7 x + 7$ have? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

End Behavior of Polynomial Functions

Date Period

Describe the end behavior of each function, or answer the question asked.

Describe the end behavior
$f (x) = x^{4} + 5 x^{3} + 7 x^{2}$
$f (x) = - 2 x^{3} - x^{2} - 4$
$f (x) = 2 x^{5} - 2 x^{4}$
$f (x) = - 5 x^{4} + 9 x - 6$
$f (x) = x^{3} + 7 x^{2} - 9 x$
$f (x) = - x^{3} + 4 x^{2} - 8 x$
$f (x) = 5 x^{4} + 6 x^{3} - 8 x - 4$
$f (x) = - 2 x^{4} - 4 x^{2}$
$f (x) = 5 x^{3} + 2 x - 1$
$f (x) = - x^{3} + 7 x^{2} + 9 x + 2$

Relate turning points and degree
A polynomial's graph has 1 turning point. What is the least possible degree?
A polynomial's graph has 5 turning points. What is the least possible degree?
At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{5} + 6 x^{4} + x^{2} + 8 x - 2$ have?
At most how many turning points can a polynomial of the same degree as $f (x) = 4 x^{6} + 9 x^{5}$ have?
At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{2} + 4$ have?
At most how many turning points can a polynomial of the same degree as $f (x) = - 4 x^{4} - 6 x^{2} + 2$ have?
A polynomial's graph has 6 turning points. What is the least possible degree?
At most how many turning points can a polynomial of the same degree as $f (x) = - 5 x^{7} + 3 x^{6} - 8 x^{5} + 6 x^{4} - 9 x^{2} - 3 x$ have?
At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{3} - 7 x$ have?
At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{2} - 7 x + 7$ have?



Sigma Prep · sigmaprep.io/worksheets Name End Behavior of Polynomial FunctionsAnswer keyVersion 1 Date Period Describe the end behavior of each function, or answer the question asked. Describe the end behavior $f (x) = x^{4} + 5 x^{3} + 7 x^{2}$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$ $f (x) = - 2 x^{3} - x^{2} - 4$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$ $f (x) = 2 x^{5} - 2 x^{4}$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$ $f (x) = - 5 x^{4} + 9 x - 6$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$ $f (x) = x^{3} + 7 x^{2} - 9 x$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$ $f (x) = - x^{3} + 4 x^{2} - 8 x$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$ $f (x) = 5 x^{4} + 6 x^{3} - 8 x - 4$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$ $f (x) = - 2 x^{4} - 4 x^{2}$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$ $f (x) = 5 x^{3} + 2 x - 1$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$ $f (x) = - x^{3} + 7 x^{2} + 9 x + 2$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$ Relate turning points and degree A polynomial's graph has 1 turning point. What is the least possible degree? $2$ A polynomial's graph has 5 turning points. What is the least possible degree? $6$ At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{5} + 6 x^{4} + x^{2} + 8 x - 2$ have? $4$ At most how many turning points can a polynomial of the same degree as $f (x) = 4 x^{6} + 9 x^{5}$ have? $5$ At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{2} + 4$ have? $1$ At most how many turning points can a polynomial of the same degree as $f (x) = - 4 x^{4} - 6 x^{2} + 2$ have? $3$ A polynomial's graph has 6 turning points. What is the least possible degree? $7$ At most how many turning points can a polynomial of the same degree as $f (x) = - 5 x^{7} + 3 x^{6} - 8 x^{5} + 6 x^{4} - 9 x^{2} - 3 x$ have? $6$ At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{3} - 7 x$ have? $2$ At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{2} - 7 x + 7$ have? $1$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

End Behavior of Polynomial FunctionsAnswer keyVersion 1

Date Period

Describe the end behavior of each function, or answer the question asked.

Describe the end behavior
$f (x) = x^{4} + 5 x^{3} + 7 x^{2}$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$
$f (x) = - 2 x^{3} - x^{2} - 4$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$
$f (x) = 2 x^{5} - 2 x^{4}$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$
$f (x) = - 5 x^{4} + 9 x - 6$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$
$f (x) = x^{3} + 7 x^{2} - 9 x$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$
$f (x) = - x^{3} + 4 x^{2} - 8 x$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$
$f (x) = 5 x^{4} + 6 x^{3} - 8 x - 4$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$
$f (x) = - 2 x^{4} - 4 x^{2}$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$
$f (x) = 5 x^{3} + 2 x - 1$ $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$
$f (x) = - x^{3} + 7 x^{2} + 9 x + 2$ $f (x) \to \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$

Relate turning points and degree
A polynomial's graph has 1 turning point. What is the least possible degree? $2$
A polynomial's graph has 5 turning points. What is the least possible degree? $6$
At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{5} + 6 x^{4} + x^{2} + 8 x - 2$ have? $4$
At most how many turning points can a polynomial of the same degree as $f (x) = 4 x^{6} + 9 x^{5}$ have? $5$
At most how many turning points can a polynomial of the same degree as $f (x) = 3 x^{2} + 4$ have? $1$
At most how many turning points can a polynomial of the same degree as $f (x) = - 4 x^{4} - 6 x^{2} + 2$ have? $3$
A polynomial's graph has 6 turning points. What is the least possible degree? $7$
At most how many turning points can a polynomial of the same degree as $f (x) = - 5 x^{7} + 3 x^{6} - 8 x^{5} + 6 x^{4} - 9 x^{2} - 3 x$ have? $6$
At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{3} - 7 x$ have? $2$
At most how many turning points can a polynomial of the same degree as $f (x) = - 3 x^{2} - 7 x + 7$ have? $1$

How to do these

$f (x) = - 3 x^{4} + 2 x^{2} - 1$

The degree is 4, which is even, so both ends agree.
The leading coefficient is negative.
So both ends go down.

The answer is $f (x) \to - \infty$ as $x \to - \infty, f (x) \to - \infty$ as $x \to \infty$ .

Where students go wrong

Reading the first term printed instead of the highest power. If the polynomial is written out of order, the leading term is not the one on the left; find the biggest exponent first.

Also called leading coefficient test, end behavior or behavior of polynomial graphs.

What you can put on this worksheet

End behavior from the equation: $f (x) = 2 x^{3} + 4 x - 6$ → $f (x) \to - \infty$ as $x \to - \infty, f (x) \to \infty$ as $x \to \infty$
Turning points and degree: At most how many turning points can a polynomial of the same degree as $f (x) = - 5 x^{5} + 9 x^{4}$ have? → $4$

Questions about these worksheets

Yes. Every end behavior sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the end behavior questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: end behavior from the equation and turning points and degree. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the first term printed instead of the highest power. If the polynomial is written out of order, the leading term is not the one on the left; find the biggest exponent first.

End behavior worksheet for Precalculus

Select your difficulty

Describe the end behavior

Relate turning points and degree

Describe the end behavior

Relate turning points and degree

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these end behavior worksheets free?+

Do the end behavior worksheets come with an answer key?+

Are these end behavior worksheets printable?+

What grade are these end behavior worksheets for?+

Can I get a different version of the same end behavior worksheet?+

Is this a free alternative to Kuta Software for end behavior?+

Can I choose how hard the end behavior questions are?+

What kinds of question are on a end behavior worksheet?+

What do students get wrong with end behavior?+