Free Printable WorksheetMathPre-CalculusMath CurriculumHSG-GPE.A.1-3(+)

Pre CAL.124 Recognize Symmetry in Parametric and Polar Representations 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.124 Recognize Symmetry in Parametric and Polar Representations. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

Pre-CAL.124 Recognize Symmetry in Parametric and Polar Representations - Primary standard selected for prompt fields in this prompt cell: CCSS HSG-GPE.A.1-3(+). Skill statement in this prompt cell: Given a parametric or polar equation, determine whether it has symmetry (and which type) and select the correct conclusion about its repeating behavior. Associated standards listed with this skill in this prompt cell: CCSS HSG-GPE.A.1-3(+); CCSS HS-Modeling; SMP 1,2,4,5,7,8; AP Precalculus CED Units 3-4; TX §111.42(c)(3)(D-H); FL MA.912.GR.7.1-7.9, MA.912.T.4.5-4.7; VA MA.AG.1-2. 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.124 Recognize Symmetry in Parametric and Polar Representations skill assessments:

Problem 1Difficulty: Mastery
A polar curve r = f(theta) has f(theta + pi/3) = f(theta), but f(-theta) is not equal to f(theta) and f(pi - theta) is not equal to f(theta). Which conclusion is justified?
AThe graph repeats every pi/3 and has rotational symmetry, including pole symmetry, but the two listed axis tests are not confirmed.
BThe graph must be symmetric about the polar axis because every periodic polar curve has that symmetry.
CThe graph must be symmetric about theta = pi/2 because pi/3 is less than pi/2.
DThe graph has no repeating behavior because the axis tests fail.
Solution & Explanation:The equality f(theta + pi/3) = f(theta) gives a pi/3 rotational repeat; after three such steps, f(theta + pi) = f(theta), so pole symmetry follows. The failed axis tests do not confirm polar-axis or theta = pi/2 symmetry.
Problem 2Difficulty: Mastery
For x = 2 + cos(2t) and y = 3 + sin(2t), which conclusion is most complete?
ASymmetric about x = 2 and y = 3; repeats every pi.
BSymmetric about the x-axis and y-axis; repeats every 2pi.
CSymmetric about the origin only; repeats every pi/2.
DNo symmetry because the center is shifted.
Solution & Explanation:This shifted circle is centered at (2, 3), so its center lines x = 2 and y = 3 are symmetries. Since the angle is 2t, the parameter period is pi.
Problem 3Difficulty: Mastery
For x = -3 + sin(8t) and y = -1 + cos(4t), choose the correct conclusion.
AHorizontal symmetry y = -1; repeats every pi/4.
BVertical symmetry x = -3; repeats every pi/2.
COrigin symmetry about (-3, -1); repeats every 2pi.
DNo symmetry because the frequencies differ.
Solution & Explanation:sin(8t) changes sign under t -> -t, so x reflects across x = -3, while cos(4t) leaves y unchanged. The shared period is pi/2.

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Frequently Asked Questions

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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 HSG-GPE.A.1-3(+) (Primary standard selected for prompt fields in this prompt cell: CCSS HSG-GPE.A.1-3(+). Skill statement in this prompt cell: Given a parametric or polar equation, determine whether it has symmetry (and which type) and select the correct conclusion about its repeating behavior. Associated standards listed with this skill in this prompt cell: CCSS HSG-GPE.A.1-3(+); CCSS HS-Modeling; SMP 1,2,4,5,7,8; AP Precalculus CED Units 3-4; TX §111.42(c)(3)(D-H); FL MA.912.GR.7.1-7.9, MA.912.T.4.5-4.7; VA MA.AG.1-2. Use this prompt-provided wording as the supporting standard description for this workflow.).

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