Free printable worksheet

Polynomial Graphs Worksheet

Twelve polynomial graphing and analysis problems involving zeros, multiplicity, degree, symmetry, turning points, and end behavior.

Algebra IIGrades 9-128 pagesAnswer key included

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What this worksheet includes

  • Twelve polynomial graphing and analysis problems involving zeros, multiplicity, degree, symmetry, turning points, and end behavior.
  • 8-page printable PDF with a complete answer key
  • Designed for Grades 9-12 algebra ii practice
  • First-page preview available before the free email unlock

How to use it

Use this polynomial graphs worksheet for guided practice, independent review, homework, tutoring sessions, or a quick skills check. The spacious layout is designed to leave room for students to show their work rather than guess from crowded answer choices.

After students finish, compare their reasoning with the included answer key. When an answer is incorrect, have the student locate the first step where their work changed direction before trying the problem again.

Who this practice is for

This resource is designed for grades 9-12 students working on algebra ii. Teachers, tutors, homeschool families, and parents can preview the first page before deciding whether the level and spacing fit the student.

Looking for a broader sequence? Browse all Algebra II worksheets.

Topic guide

How to approach polynomial graphs

Polynomial Graphs practice is most effective when students identify the governing definition, theorem, or rule before beginning a calculation. Twelve polynomial graphing and analysis problems involving zeros, multiplicity, degree, symmetry, turning points, and end behavior.

This algebra ii set moves from direct applications into mixed review. Students should preserve exact values, label each important step, and use the answer key to locate the first point where an incorrect solution changes direction.

Learning objectives

  • Recognize the central rules and representations used in polynomial graphs
  • Apply an efficient procedure to solve polynomial graphs problems
  • Check results using notation, substitution, estimation, or a relevant inverse process

Skills to review first

  • Core vocabulary connected to polynomial graphs
  • Accurate arithmetic and algebraic simplification
  • Reading mathematical notation and showing work in a logical sequence
Worked example

Follow the reasoning step by step

Plan and solve a representative polynomial graphs problem from Part A.

  1. List the given information and state what must be found.
  2. Select the definition, theorem, formula, or procedure that applies to polynomial graphs.
  3. Substitute or transform carefully, keeping one logical change on each line.
  4. Check the result against the original conditions and include units or restrictions when needed.

Solution: A complete solution includes the correct result and enough reasoning to verify it.

Try before downloading

Three sample problems

Work each problem, then open the answer to check your result.

Problem 1 · foundations

Analyze the graph: List the xx-intercepts. At each, does the graph cross or touch?

Polynomial curve for identifying intercepts and multiplicityxy-4-3-2-11234-4-224
Polynomial curve for identifying intercepts and multiplicity
Show answer

x=−2 touches; x=2 crossesx=-2\text{ touches};\ x=2\text{ crosses}

Problem 2 · skill practice

Sketch on the coordinate plane: f(x)=0.45(x+2)x(x−2)f(x)=0.45(x+2)x(x-2)

Blank coordinate plane for sketching the polynomialxy-4-3-2-11234-4-224
Blank coordinate plane for sketching the polynomial
Show answer

x=−2,0,2; left down, right upx=-2,0,2;\ \text{left down, right up}

One correct sketch with zeros at negative two, zero, and twoxy-4-3-2-11234-4-224
One correct sketch with zeros at negative two, zero, and two
Problem 3 · mixed review

Construct a polynomial graph: Zero −1 has multiplicity 2; zero 3 has multiplicity 1; negative leading coefficient.

Blank coordinate plane for constructing the polynomialxy-4-3-2-11234-4-224
Blank coordinate plane for constructing the polynomial
Show answer

Sample: f(x)=−0.18(x+1)2(x−3)\text{Sample: }f(x)=-0.18(x+1)^2(x-3)

One valid polynomial with a double zero at negative one and a simple zero at threexy-4-3-2-11234-4-224
One valid polynomial with a double zero at negative one and a simple zero at three
Error check

Common mistakes to watch for

  • Beginning a calculation before identifying the relevant rule or given information
  • Dropping a sign, unit, restriction, or exact-value symbol during simplification
  • Comparing only the final answer instead of checking the first incorrect step
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