Free Printable WorksheetMathPre-CalculusMath CurriculumHSF-TF.1-9(+)

Pre CAL.104 Interpret Symmetry and Periodicity in Polar Graphs Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Pre-Calculus Math worksheet provides structured practice on Pre CAL.104 Interpret Symmetry and Periodicity in Polar Graphs. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

Pre-CAL.104 Interpret Symmetry and Periodicity in Polar Graphs - Primary standard selected for prompt fields in this prompt cell: CCSS HSF-TF.1-9(+). Skill statement in this prompt cell: Given a polar equation, determine which symmetry test applies and select the correct conclusion about symmetry and periodicity of the graph. Associated standards listed with this skill in this prompt cell: CCSS HSF-TF.1-9(+); CCSS HSF-IF.7e; CCSS HSG-SRT.C.6-8; SMP 1,2,3,5,7,8; AP Precalculus CED Unit 3; TX §111.42(c)(4)(A-D), (c)(5)(I); FL MA.912.T.1.3-1.8, MA.912.T.2.1-2.5, MA.912.T.3.1-3.3, MA.912.T.4.1-4.7; VA T.CT.1-2, T.GT.1-2, T.IE.1-3. Use this prompt-provided wording as the supporting standard description for this workflow.

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Pre-Calculus

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related Pre CAL.104 Interpret Symmetry and Periodicity in Polar Graphs skill assessments:

Problem 1Difficulty: Mastery

For the equation r = 1 + cos(4theta) + sin(6theta) + cos(8theta), choose the direct-test conclusion that avoids overclaiming symmetry.

AFor r = 1 + cos(4theta) + sin(6theta) + cos(8theta), the complete direct-test result is polar-axis symmetry only (theta -> -theta), and the equation repeats every pi/2 in theta.

BFor r = 1 + cos(4theta) + sin(6theta) + cos(8theta), the complete direct-test result is symmetry about line theta = pi/2 only (theta -> pi - theta), and the equation repeats every pi/2 in theta.

CFor r = 1 + cos(4theta) + sin(6theta) + cos(8theta), the complete direct-test result is polar-axis, line theta = pi/2, and pole symmetry by theta substitutions, and the equation repeats every pi/2 in theta.

DFor r = 1 + cos(4theta) + sin(6theta) + cos(8theta), the complete direct-test result is pole symmetry only (theta -> theta + pi), and the equation repeats every pi in theta.

Solution & Explanation:theta -> theta + pi leaves cos(4theta), sin(6theta), and cos(8theta) unchanged. The periods pi/2, pi/3, and pi/4 have common repeat pi. Use these direct tests only as confirming tests, so choose the option that states the complete direct-test result and repeat interval.

Problem 2Difficulty: Mastery

Which complete conclusion about r = 2cos(3theta)sin(theta) is supported before graphing the polar curve?

AFor r = 2cos(3theta)sin(theta), the complete direct-test result is pole symmetry only (theta -> theta + pi), and the equation repeats every 4pi in theta.

BFor r = 2cos(3theta)sin(theta), the complete direct-test result is polar-axis symmetry only (theta -> -theta), and the equation repeats every 4pi in theta.

CFor r = 2cos(3theta)sin(theta), the complete direct-test result is pole symmetry only (theta -> theta + pi), and the equation repeats every pi in theta.

DFor r = 2cos(3theta)sin(theta), the complete direct-test result is symmetry about line theta = pi/2 only (theta -> pi - theta), and the equation repeats every 4pi in theta.

Solution & Explanation:Under theta -> theta + pi, both factors change sign, keeping the product the same. The product simplifies to sin(4theta) - sin(2theta), whose common repeat is pi. Use these direct tests only as confirming tests, so choose the option that states the complete direct-test result and repeat interval.

Problem 3Difficulty: Mastery

A graphing plan starts with symmetry and period for r = 2sin(3theta)cos(theta). Which statement should be used?

AFor r = 2sin(3theta)cos(theta), the complete direct-test result is pole symmetry only (theta -> theta + pi), and the equation repeats every 2pi in theta.

BFor r = 2sin(3theta)cos(theta), the complete direct-test result is pole symmetry only (theta -> theta + pi), and the equation repeats every pi in theta.

CFor r = 2sin(3theta)cos(theta), the complete direct-test result is polar-axis symmetry only (theta -> -theta), and the equation repeats every 2pi in theta.

DFor r = 2sin(3theta)cos(theta), the complete direct-test result is symmetry about line theta = pi/2 only (theta -> pi - theta), and the equation repeats every 2pi in theta.

Solution & Explanation:Adding pi to theta changes both sin(3theta) and cos(theta) signs, so the product stays unchanged. The product simplifies to sin(4theta) + sin(2theta), which repeats every pi. Use these direct tests only as confirming tests, so choose the option that states the complete direct-test result and repeat interval.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Pre-Calculus Math students. It aligns with educational standard HSF-TF.1-9(+) (Primary standard selected for prompt fields in this prompt cell: CCSS HSF-TF.1-9(+). Skill statement in this prompt cell: Given a polar equation, determine which symmetry test applies and select the correct conclusion about symmetry and periodicity of the graph. Associated standards listed with this skill in this prompt cell: CCSS HSF-TF.1-9(+); CCSS HSF-IF.7e; CCSS HSG-SRT.C.6-8; SMP 1,2,3,5,7,8; AP Precalculus CED Unit 3; TX §111.42(c)(4)(A-D), (c)(5)(I); FL MA.912.T.1.3-1.8, MA.912.T.2.1-2.5, MA.912.T.3.1-3.3, MA.912.T.4.1-4.7; VA T.CT.1-2, T.GT.1-2, T.IE.1-3. Use this prompt-provided wording as the supporting standard description for this workflow.).