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Pre-calculus · AP

AP-aligned materials for 1.9 (Polynomial and Rational Functions). This topic is assessed on the AP Exam.

What this standard means

  • Rational Functions and Vertical Asymptotes Determine holes in graphs of rational functions. If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, then the graph of the rational function has a hole at the corresponding input value. 1.10.A.2 If the graph of a rational function r has a hole at x = c, then the location of the hole can be determined by examining the output values corresponding to input values sufficiently close to c. If input values sufficiently close to c correspond to output values arbitrarily close to L, then the hole is located at the point with coordinates (c L, ). The corresponding mathematical notation is lim r x( ) = L x c . It should be noted that lim r x( ) = lim r x( ) = lim r x( ) = L. − + x c x c x c
  • Demonstrate mastery on AP Exam-style questions for this topic.

_See printable PDF for diagram._

How to use the 20 practice sets

| Sets | When to use | | --- | --- | | 1–5 | Intro — explore together, short written items | | 6–10 | Core skills — diagrams and written practice | | 11–15 | Mixed review — explain thinking | | 16–20 | Stretch — word problems and mastery tasks |

Pacing: 10–15 minutes per session.

How to practice

1. Label unit circle coordinates for key angles 2. Connect radians and degrees 3. Graph sine and cosine with amplitude and period

_See printable PDF for diagram._

Common mistakes

  • Wrong quadrant signs
  • Period or amplitude errors in graphs

Review and practice tests

1. Start Review 1/10 when sets 1–3 feel comfortable. 2. Move up one review level with little help. 3. Use Practice Test 4/10–6/10 for mid-standard checks. 4. Practice Test 10/10 is the mastery bar for 1.9.

  • [ ] Evaluates trig functions at standard angles
  • [ ] Uses unit circle reasoning
  • [ ] Graphs basic trig functions with transformations

Materials for this standard

  • Practice Problems — 20 printable sets
  • Review — 10 difficulty levels
  • Practice Test — 10 difficulty levels
  • Answer key — for parents and tutors

For your personal study only. Copying, printing, screenshotting, or sharing this material is not permitted and may result in loss of access.

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