Worksheets · Precalculus

Precalculus graphing polynomial functions worksheet

Precalculus graphs polynomials given in standard form, so factoring comes first. Multiplicity decides the behavior at each zero: an odd one crosses the axis and an even one touches and turns back.

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

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Pick more than one for a sheet that mixes them.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing Polynomial Functions

Date Period

Graph each function.

  1. Graph y=−(x+2)(x−1)(x−2)-10-10-5-500551010
  2. Graph y=−x3−x2+4x+4-10-10-5-500551010
  3. Graph y=−(x+3)(x+2)(x−1)-10-10-5-500551010
  4. Graph y=−x3+2x2+3x-10-10-5-500551010
  5. Graph y=(x+1)x(x−1)-10-10-5-500551010
  6. Graph y=x3−x-10-10-5-500551010
  7. Graph y=(x+3)(x+1)(x−1)-10-10-5-500551010
  8. Graph y=−x3+x-10-10-5-500551010
  9. Graph y=−(x+3)(x+1)x-10-10-5-500551010
  10. Graph y=x3−x2−4x+4-10-10-5-500551010
  11. Graph y=−x(x−1)(x−3)-10-10-5-500551010
  12. Graph y=x3+2x2−x−2-10-10-5-500551010
  13. Graph y=−(x+1)x(x−1)-10-10-5-500551010
  14. Graph y=x3+2x2−3x-10-10-5-500551010
  15. Graph y=(x+1)(x−2)(x−3)-10-10-5-500551010
  16. Graph y=−x3+x2+2x-10-10-5-500551010
  17. Graph y=(x+2)(x+1)x-10-10-5-500551010
  18. Graph y=−x3+2x2+5x−6-10-10-5-500551010
  19. Graph y=x(x−2)(x−3)-10-10-5-500551010
  20. Graph y=x3−x2−6x-10-10-5-500551010

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing Polynomial FunctionsAnswer keyVersion 1

Date Period

Graph each function.

  1. Graph y=−(x+2)(x−1)(x−2)-10-10-5-500551010Zeros −2,1,2, ends up then down
  2. Graph y=−x3−x2+4x+4-10-10-5-500551010zeros −2,−1,2; rises to the left, falls to the right
  3. Graph y=−(x+3)(x+2)(x−1)-10-10-5-500551010Zeros −3,−2,1, ends up then down
  4. Graph y=−x3+2x2+3x-10-10-5-500551010zeros −1,0,3; rises to the left, falls to the right
  5. Graph y=(x+1)x(x−1)-10-10-5-500551010Zeros −1,0,1, ends down then up
  6. Graph y=x3−x-10-10-5-500551010zeros −1,0,1; falls to the left, rises to the right
  7. Graph y=(x+3)(x+1)(x−1)-10-10-5-500551010Zeros −3,−1,1, ends down then up
  8. Graph y=−x3+x-10-10-5-500551010zeros −1,0,1; rises to the left, falls to the right
  9. Graph y=−(x+3)(x+1)x-10-10-5-500551010Zeros −3,−1,0, ends up then down
  10. Graph y=x3−x2−4x+4-10-10-5-500551010zeros −2,1,2; falls to the left, rises to the right
  11. Graph y=−x(x−1)(x−3)-10-10-5-500551010Zeros 0,1,3, ends up then down
  12. Graph y=x3+2x2−x−2-10-10-5-500551010zeros −2,−1,1; falls to the left, rises to the right
  13. Graph y=−(x+1)x(x−1)-10-10-5-500551010Zeros −1,0,1, ends up then down
  14. Graph y=x3+2x2−3x-10-10-5-500551010zeros −3,0,1; falls to the left, rises to the right
  15. Graph y=(x+1)(x−2)(x−3)-10-10-5-500551010Zeros −1,2,3, ends down then up
  16. Graph y=−x3+x2+2x-10-10-5-500551010zeros −1,0,2; rises to the left, falls to the right
  17. Graph y=(x+2)(x+1)x-10-10-5-500551010Zeros −2,−1,0, ends down then up
  18. Graph y=−x3+2x2+5x−6-10-10-5-500551010zeros −2,1,3; rises to the left, falls to the right
  19. Graph y=x(x−2)(x−3)-10-10-5-500551010Zeros 0,2,3, ends down then up
  20. Graph y=x3−x2−6x-10-10-5-500551010zeros −2,0,3; falls to the left, rises to the right

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Describe the graph of f(x)=−x3+4x2−4x

  1. Factor: f(x) = -x(x² - 4x + 4) = -x(x - 2)².
  2. x = 0 has multiplicity 1, so it crosses. x = 2 has multiplicity 2, so it touches.
  3. Degree 3 with a negative lead: up on the left, down on the right.

The answer is Crosses at 0, touches at 2, rises left, falls right.

Where students go wrong

Crossing the axis at a squared factor. A zero of even multiplicity does not change the sign of f, so the graph touches the axis there and turns back.

Also called polynomial graphs, end behavior graphs or plotting polynomials from factors.

What you can put on this worksheet

A polynomial from its factors
Graph y=−x(x−2) → Zeros 0,2, ends both down
From standard form, factored first
Graph y=−x3−x2+6x → zeros −3,0,2; rises to the left, falls to the right
Calculator zeros and extrema to the nearest tenth
Use a graphing calculator to find the relative maxima and minima to the nearest tenth: f(x)=x4+x3−5x+3 → Maximum: noneMinimum: (0.9,−0.1)

Questions about these worksheets

Yes. Every graphing polynomial functions 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 graphing polynomial functions 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 3: a polynomial from its factors, from standard form, factored first and calculator zeros and extrema to the nearest tenth. Tick as many as you want and set how many of each, or let it spread them evenly.

Getting the ends the wrong way round. An odd number of factors sends the two ends opposite ways and an even number sends them the same way, with the leading sign deciding which. The other slip is forgetting the curve has to turn between consecutive zeros.