Free Printable WorksheetMathAlgebra IIMath CurriculumF-IF.7d

ALG2.135 Identifying Holes Versus Asymptotes Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Algebra II Math worksheet provides structured practice on ALG2.135 Identifying Holes Versus Asymptotes. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

ALG2.135 Identifying Holes Versus Asymptotes - Primary standard selected for prompt fields in this prompt cell: CCSS F-IF.7d. Skill statement in this prompt cell: Identify whether a discontinuity is removable (a hole) or a vertical asymptote by analyzing common factors and domain restrictions. Associated standards listed with this skill in this prompt cell: CCSS F-IF.7d; CA CCSSM F-IF.7d; MA 2017 F-IF.7d; NY NGLS F-IF.7d; TX TEKS Algebra II: §111.40. 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

Algebra II

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 ALG2.135 Identifying Holes Versus Asymptotes skill assessments:

Problem 1Difficulty: Mastery

Use common factors and denominator restrictions to classify the discontinuities of f(x) = [(x - 4)² (x + 1) (x - 6)]/[(x - 4)² (x + 1) (x - 2)].

AHoles: x = -1 and x = 4; vertical asymptotes: x = 2

BHoles: none; vertical asymptotes: x = -1, x = 2, and x = 4

CHoles: x = -1, x = 2, and x = 4; vertical asymptotes: none

DHoles: x = 2; vertical asymptotes: x = -1 and x = 4

Solution & Explanation:Compare factors in the numerator and denominator. Denominator factors that cancel completely give holes at x = -1 and x = 4; denominator factors that remain give vertical asymptotes at x = 2.

Problem 2Difficulty: Mastery

Which conclusion correctly combines simplification and discontinuity classification for f(x) = [(x + 2)² (x - 3) (x - 5)]/[(x + 2) (x - 3) (x - 5)² (x + 7)]?

ASimplified expression: [x + 2]/[x² + 2x - 35]; Holes: none; vertical asymptotes: x = -7, x = -2, x = 3, and x = 5

BSimplified expression: [x + 2]/[x² + 2x - 35]; Holes: x = -7, x = -2, x = 3, and x = 5; vertical asymptotes: none

CSimplified expression: [x + 2]/[x² + 2x - 35]; Holes: x = -7 and x = 5; vertical asymptotes: x = -2 and x = 3

DSimplified expression: [x + 2]/[x² + 2x - 35]; Holes: x = -2 and x = 3; vertical asymptotes: x = -7 and x = 5

Solution & Explanation:Simplifying gives [x + 2]/[x² + 2x - 35], but the original denominator restrictions still matter. Restrictions from factors that cancel completely are holes at x = -2 and x = 3; restrictions from factors that remain are vertical asymptotes at x = -7 and x = 5.

Problem 3Difficulty: Mastery

Which factor cancels completely and creates a removable discontinuity in f(x) = [(x - 3)(x + 4)(x - 8)]/[(x - 3)(x + 4)²(x + 1)]?

A(x + 9)

B(x + 4)

C(x - 3)

D(x + 1)

Solution & Explanation:The factor (x - 3) appears in the denominator and is canceled completely by the numerator. Its original domain restriction stays, so it creates a hole.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Algebra II Math students. It aligns with educational standard F-IF.7d (Primary standard selected for prompt fields in this prompt cell: CCSS F-IF.7d. Skill statement in this prompt cell: Identify whether a discontinuity is removable (a hole) or a vertical asymptote by analyzing common factors and domain restrictions. Associated standards listed with this skill in this prompt cell: CCSS F-IF.7d; CA CCSSM F-IF.7d; MA 2017 F-IF.7d; NY NGLS F-IF.7d; TX TEKS Algebra II: §111.40. Use this prompt-provided wording as the supporting standard description for this workflow.).