Algebra II · Polynomial functions
Students match one polynomial across eight views: equation, factored form, graph, zeros, multiplicity, end behavior, degree, and leading coefficient. The free pack includes three levels, printable cards, a no-cut worksheet, teacher directions, and complete answer keys.
All 14 function families were recalculated and every graph was checked against its equation before publication. Free to print; no signup or email required.
Browse our collection of free polynomial worksheets and answer keys for operations, factoring, division, graphing, and zeros.
Choose cut-out cards for stations and partner work, or use the no-cut version when students need a conventional worksheet. The teacher key covers both formats.
All three levels, category-labeled cards, dashed cut lines, and a recording sheet before each set.
Graphs labeled by letter with independently scrambled options. Students write the matching graph letter in each box.
Setup, pacing, differentiation, standards, cut-card code matches, and the numbered answers for the no-cut version.
The printable cards, uncut worksheets, teacher directions, and all three answer keys in one file.
A match is only complete when every representation tells the same story. The category colors help students organize a large set; they do not identify which cards belong together.
The number of card types stays the same. What changes is the amount of structure students must track at once.
Four functions with integer zeros and visually clear crosses and touches.
Five functions with mixed multiplicities and positive or negative leading coefficients.
Five functions that require students to distinguish repeated roots with similar-looking local behavior.
This walkthrough uses a function that is not in any of the three printable levels, so nothing here gives away a card. The three marked points are the evidence a student should be able to point to.
The sort assumes a few skills are already there. If they aren't, students match by shape alone and the activity stops teaching anything.
Factoring a quadratic and a simple cubic, reading an x-intercept off a graph, and knowing that a factor (x - r) gives you the zero x = r. If factoring is still shaky, start with the factoring polynomials flowchart guide and come back to this.
That one feature is never enough. Three of the five standard functions have degree 4, and two of those have identical end behavior, so stopping at the first plausible card puts you in the wrong family and you have to back up and find a second piece of evidence.
Place the eight category stacks face up. Partners build one family at a time and cannot place a match until both students can name the evidence.
Use one level at each station. At the final station, students choose one disputed match and write a short defense using multiplicity or end behavior.
Give students the graph-letter version and the two written justifications. It needs no scissors, envelopes, or advance sorting.
Students don't check their work against end behavior. They spot two intercepts, grab the first card showing those zeros, and move on. So when a match goes wrong, and it will, I have them test a point: pick one or two x-values between the turns, put them in the equation, and see where the point lands. If it isn't anywhere near the curve on the graph card, something in that row is wrong, and they've found it themselves instead of waiting on me or on the key. That routine is printed on every recording sheet and worksheet in the download.
The challenge set has a fifth-degree function showing only two x-intercepts. Count intercepts, call it a quadratic, and nothing about the tails works after that.
This is the one that costs the most time. In the standard set, x = -1 is a triple root of (x + 1)³(x - 2). The graph flattens right out as it passes through, so it reads as a touch, and the even-multiplicity card goes down next to it. Odd multiplicity always crosses, however flat it gets on the way.
Parity tells you whether the two tails match. It says nothing about which way they point — that's the leading sign. Every level carries a positive and a negative version of the same degree for exactly this reason.
Leading coefficients of ±1, ±2, and +1/2 are scattered through the set and none of them change the shape. Short of a test point, there's no way to tell 2(x + 3)(x - 1)² from (x + 3)(x - 1)².
The full key for all three levels is in the teacher PDF. This is the introductory set, so you can see the format before you send 31 pages to the copier.
| Graph | Expanded equation | Factored form | Zeros | Multiplicity | End behavior | Degree | Leading coeff. |
|---|---|---|---|---|---|---|---|
| A | y = x² + x - 2 | y = (x + 2)(x - 1) | {-2, 1} | both m = 1 | left up; right up | 2 | +1 |
| B | y = -x² + 2x - 1 | y = -(x - 1)² | {1} | x = 1: m = 2 | left down; right down | 2 | -1 |
| C | y = x³ - 3x - 2 | y = (x + 1)²(x - 2) | {-1, 2} | x = -1: m = 2; x = 2: m = 1 | left down; right up | 3 | +1 |
| D | y = -x³ + 2x² + 5x - 6 | y = -(x + 2)(x - 1)(x - 3) | {-2, 1, 3} | all m = 1 | left up; right down | 3 | -1 |
Rows A and B share a degree, and rows A and C share a leading coefficient, so those two cards are interchangeable between the rows that use them.
Aligned to the Common Core algebra standards for polynomials and, for California classrooms, to A-APR.3 as listed by the California Department of Education.
| Skill | What students use | CCSS connection | What a mismatch usually means |
|---|---|---|---|
| Zeros and factors | Every zero x=r corresponds to a factor (x-r). | HSA.APR.B.3; HSA.SSE.A.2 | The student may be copying signs instead of solving each factor equation. |
| Multiplicity | Odd multiplicity crosses; even multiplicity touches and turns. | HSA.APR.B.3 | The student sees the intercept but not how the graph behaves there. |
| Degree and tails | Even degree has matching tail directions; odd degree has opposite directions. | HSF.IF.C.7c | The student may know the degree but not connect parity to a graph. |
| Leading coefficient | The sign determines the right-tail direction once degree parity is known. | HSF.IF.C.7c | The student may be treating degree alone as enough to determine end behavior. |
| Multiple representations | Expanded, factored, graphical, and verbal descriptions must agree. | MP2; MP3 | The student may be pattern-matching from one feature instead of checking the whole structure. |
This is a classroom-made, criterion-referenced activity, not a standardized or norm-referenced assessment. Use the pattern of mismatches and the written explanations to plan reteaching.
Teachers and families may print, copy, and adapt these files for their own students at no cost. Please link to this guide rather than reposting the PDFs, so the current answer key and any correction stay with the activity.
Citing this activity
Sahbazoglu, B. (2026). Polynomial graph-matching worksheet and card sort [Classroom activity]. Burke Tutoring in Fremont. https://burketutoringinfremont.com/polynomial-graph-card-sort/
Related polynomial practice: start with the polynomial skills diagnostic test, review structure with the factoring polynomials flowchart guide, follow the sort with the polynomial error analysis worksheet, or browse every free quadratics worksheet and answer key.
It is a classification activity in which students match several representations of the same polynomial function. This set uses eight: expanded equation, factored form, graph, zeros, multiplicity, end behavior, degree, and leading coefficient.
Either can work, but the strongest routine moves back and forth. Students can read zeros and cross/touch behavior from the graph, then use the factors to confirm the zeros and multiplicities. Degree and leading sign provide a separate check on the tails.
Yes. The uncut worksheet shows graphs labeled by letter and lists the other representations in a different order. Students write the matching graph letter in each answer box.
The introductory set usually fits in 20-30 minutes. The standard and challenge sets generally take 30-50 minutes when students are required to explain at least two matches.
No. Color identifies the representation category only. Every expanded equation has the same category color, for example, while the card codes and option numbers are independently scrambled.
That is intentional. Identical property cards are mathematically interchangeable, so either code or option order is correct as long as each copy is used once. The teacher key presents one valid arrangement.
Level 1 fits students who have just met multiplicity, level 2 fits a standard Algebra 2 unit review, and level 3 fits honors classes or precalculus review. Most classes can run level 1 as a warm-up and level 2 the same period.
You may print and share it directly with your own students. For a public website, resource list, or blog post, link to this page instead of reposting the PDF so visitors receive the current version and correction log.
August 5, 2026: Rebuilt all four PDFs and the decision map. Corrected the introductory no-cut worksheet, which asked for graph letters A-E on a level that only has graphs A-D, enlarged every worksheet graph, moved the answer keys to landscape so all eight columns are readable, and added the test-point checking routine to the recording sheets, the worksheets, the teacher guide, and the decision map. On this page: replaced the sample function so it no longer duplicates an introductory card, rebuilt the sample graph so the curve runs off the frame instead of flattening at the top and bottom, and added a prerequisites section, an answer-key preview, and a citation block.
August 4, 2026: Published the three-level cut-out activity, no-cut worksheet, teacher key, complete bundle, and embeddable graph-matching decision map.
Accuracy review: Recomputed all 14 expanded forms from their factors, confirmed the zeros and multiplicities of every function, checked every degree, leading coefficient, and end-behavior match against the key, and confirmed that each card code and each numbered no-cut option is used exactly once per level.
Written and reviewed by
Berke is the founder and a tutor at Burke Tutoring in Fremont. He has tutored math and science for more than 10 years and develops Algebra I and Algebra II practice activities from problem patterns he has used with students in Fremont, Newark, and Union City.
Read Berke’s full tutor profile →Burke Tutoring provides in-home math tutoring in Fremont, Newark, and Union City, including polynomial functions, factoring, graphing, and test review. Call (510) 453-0350 to talk through what your student needs.
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