--- title: "AP Precalculus Unit 3 Practice Test" description: "A timed AP Precalculus Unit 3 (Trigonometric and Polar Functions) practice test in the AP format, drawn from 1,327 multiple-choice questions and 20 FRQs written for the unit. The standard version takes about 85 minutes." canonical: "https://fiveable.me/ap-pre-calc/unit-test/3" type: "unit-test" subject: "AP Precalculus" unit: "Unit 3" --- # AP Precalculus Unit 3 Practice Test ## Overview A timed AP Precalculus Unit 3 (Trigonometric and Polar Functions) practice test in the AP format, drawn from 1,327 multiple-choice questions and 20 FRQs written for the unit. The standard version takes about 85 minutes. ## Test Format - 20 Multiple Choice Questions (50 minutes) - 1 FRQ – FRQ 3 – Modeling a Periodic Context (No Calculator) (18 minutes) - 1 FRQ – FRQ 4 – Symbolic Manipulations (No Calculator) (18 minutes) ## Unit FRQs by Type - [FRQ 3 – Modeling a Periodic Context (No Calculator)](/ap-pre-calc/frq-practice?type=frq-3-modeling-a-periodic-context-no-calculator&id=6977caba384a204905d878e8): 10 questions - [FRQ 4 – Symbolic Manipulations (No Calculator)](/ap-pre-calc/frq-practice?type=frq-4-symbolic-manipulations-no-calculator&id=6a3d8970f456441fd3b79b09): 10 questions ## Practice Questions ### Question 1 A sound wave is modeled by a periodic function where successive peaks (maximum values) occur at times t = 1.2 seconds and t = 3.6 seconds. Which statement correctly identifies the period of this sound wave and explains the reasoning? - (A) The period is 2.4 seconds because the distance between successive peaks is one complete cycle, and 3.6 − 1.2 = 2.4. - (B) The period is 1.2 seconds because the first peak occurs at t = 1.2 seconds. - (C) The period is 4.8 seconds because two complete cycles occur between t = 1.2 and t = 3.6 seconds. - (D) The period cannot be determined from peak times alone; we need to know the times of both peaks and troughs to calculate the period. **Answer: (A)** Recognizes that successive peaks are separated by exactly one period. Since peaks represent the same characteristic (local maximum) occurring at consecutive cycles, the time between them is the period. The calculation 3.6 − 1.2 = 2.4 correctly identifies this interval. ### Question 2 A periodic function is used to model the depth of water in a tidal pool. Measurements show that the water depth reaches its maximum of 6 feet at 2:00 PM and again at 2:50 PM. Which statement correctly describes the period of this tidal function and explains the reasoning? - (A) The period is 50 minutes because the time between successive maxima is one complete cycle, and the maximum repeats every 50 minutes. - (B) The period is 25 minutes because the maximum occurs at the midpoint of each cycle. - (C) The period is 100 minutes because two complete cycles must occur between the first and second maximum. - (D) The period cannot be determined from maximum times alone; we also need to know when the water depth reaches its minimum. **Answer: (A)** Recognizes that successive maxima are separated by exactly one period. Since maxima represent the same characteristic (local maximum) occurring at consecutive cycles, the time interval between them equals the period. The 50-minute interval from 2:00 PM to 2:50 PM represents one complete cycle. ### Question 3 A buoy in the ocean oscillates up and down due to wave motion. An observer records that the buoy returns to the same height and is moving in the same direction every 8 seconds. Which statement correctly describes the period of this periodic phenomenon? - (A) The period is 8 seconds; position and velocity both repeat. - (B) The period is 4 seconds; maximum height marks half a cycle. - (C) The period is 16 seconds; two oscillations define true repetition. - (D) The period cannot be determined without the maximum and minimum heights. **Answer: (A)** Period is the smallest positive time interval for complete repetition of both position and velocity. The observation confirms one full cycle occurs every 8 seconds, so the period is 8 seconds. ### Question 4 A periodic function g(x) has period 6. If g(2) = 5 and g(2) is a local maximum, which statement correctly describes g(8) and explains why? - (A) g(8) = 5 and is a local maximum because periodic functions repeat all characteristics. - (B) g(8) = 5, but is a local minimum because characteristics alternate between periods. - (C) g(8) cannot be determined because we lack the complete formula for g(x). - (D) g(8) = 5, but we cannot determine if it is a maximum or minimum without more information. **Answer: (A)** Since the period is 6, g(8) = g(2 + 6) = g(2) = 5 by the definition of periodicity. Periodic functions repeat all structural characteristics—including local extrema—at corresponding points in each period, so g(8) must be a local maximum. ### Question 5 A periodic function h(x) has period 5. Over [0, 5), it increases from x = 0 to x = 1.5, decreases from x = 1.5 to x = 4, and increases from x = 4 to x = 5. Which statement correctly describes h(x) over [5, 10)? - (A) h(x) increases on [5, 6.5], decreases on [6.5, 9], and increases on [9, 10]. - (B) h(x) decreases on [5, 6.5], increases on [6.5, 9], and decreases on [9, 10]. - (C) h(x) increases on [5, 7.5], decreases on [7.5, 9], and increases on [9, 10]. - (D) The behavior cannot be determined without specific function values in [5, 10). **Answer: (A)** Periodic functions repeat all characteristics identically in each period. Since [5, 10) is the second period, the increase/decrease pattern shifts by 5 but remains unchanged: [0, 1.5] → [5, 6.5], [1.5, 4] → [6.5, 9], [4, 5] → [9, 10]. ### Question 6 A water treatment facility monitors chlorine levels in a storage tank. The facility records that chlorine concentration reaches 8 ppm at noon, returns to 8 ppm at 6 PM, reaches a minimum of 2 ppm at 3 PM, and completes this pattern every 12 hours. Which statement correctly verifies that this relationship is periodic and identifies its period? - (A) The relationship is periodic with period 12 hours because the pattern repeats every 12 hours. - (B) The relationship is periodic with period 6 hours because the minimum occurs at the midpoint. - (C) The relationship is periodic with period 6 hours because concentration drops from 8 ppm to 2 ppm. - (D) The relationship is not periodic because concentration does not return to 8 ppm every 6 hours. **Answer: (A)** Periodicity requires the same pattern to repeat over successive equal-length intervals. The chlorine concentration follows the same cycle every 12 hours, which defines a periodic relationship with period 12 hours. ### Question 7 A Ferris wheel has a diameter of 40 meters with its center 22 meters above ground. A rider's height above ground is 22 meters when boarding at the bottom, reaches 62 meters at the top, and returns to 22 meters after one complete revolution. The height function repeats this pattern for every complete revolution. Which statement correctly verifies that the height function is periodic and describes its period? - (A) The function is periodic with period one complete revolution because the height pattern repeats identically. - (B) The function is periodic with period half a revolution because maximum height occurs at the midpoint. - (C) The function is periodic with period one revolution only if the rider boards at the bottom. - (D) The function is not periodic because height values change continuously rather than repeating discretely. **Answer: (A)** Periodicity is verified by confirming that the output pattern repeats over successive equal-length intervals. Each complete revolution represents one period because the rider returns to the same height and follows the same path. ### Question 8 A sound wave has amplitude values measured at equal time intervals: 0 ms → 0, 2.5 ms → 5 units, 5 ms → 0, 7.5 ms → −5 units, 10 ms → 0. This pattern repeats identically. Which statement correctly verifies periodicity and determines the period? - (A) Periodic with period 10 ms; the complete pattern repeats every 10 ms. - (B) Periodic with period 5 ms because amplitude returns to zero at 5 ms and 10 ms. - (C) Periodic with period 2.5 ms because amplitude changes by 5 units at each interval. - (D) Not periodic because amplitude takes both positive and negative values throughout. **Answer: (A)** Periodicity is verified by observing that the complete amplitude sequence repeats identically over equal-length intervals. The period is 10 ms because this is the smallest interval after which the entire pattern repeats. ### Question 9 A pendulum swings back and forth with the following motion: at time 0 seconds it is at center position (0 cm), at 0.5 seconds it reaches maximum displacement (10 cm), at 1 second it returns to center (0 cm), at 1.5 seconds it reaches maximum displacement in the opposite direction (−10 cm), and at 2 seconds it returns to center (0 cm). This pattern repeats identically every 2 seconds. Which statement correctly verifies that the pendulum's displacement is a periodic function and identifies its period? - (A) The displacement is periodic with period 2 seconds because the complete pattern repeats identically. - (B) The displacement is periodic with period 1 second because displacement returns to 0 cm. - (C) The displacement is periodic with period 0.5 seconds because maximum displacement occurs every 0.5 seconds. - (D) The displacement is not periodic because maximum and minimum displacements have opposite signs. **Answer: (A)** Periodicity is verified by confirming that the complete sequence of displacement values repeats identically over equal-length time intervals. The period is 2 seconds because this is the smallest interval over which the entire pattern repeats. ### Question 10 A tidal phenomenon shows high tide at 6:00 AM (8 feet), low tide at 12:30 PM (2 feet), and high tide again at 7:00 PM (8 feet). Which statement correctly verifies that this pattern is periodic and identifies its period? - (A) The pattern is periodic with a period of 13 hours - (B) The pattern is periodic with a period of 6.5 hours - (C) The pattern is not periodic because water level changes continuously - (D) The pattern is periodic with a period of 12 hours **Answer: (A)** Periodicity requires the function to return to the same output value at regular intervals. The water level of 8 feet occurs at 6:00 AM and 7:00 PM, a 13-hour interval, satisfying the definition that period k is the smallest positive value where f(x + k) = f(x). ## Review the Unit - [3.1 Periodic Phenomena](/ap-pre-calc/unit-3/periodic-phenomena/study-guide/xef2FVxbcWiHTFgh) - [3.2 Sine, Cosine, and Tangent](/ap-pre-calc/unit-3/sine-cosine-tangent/study-guide/6r53DIMsbdFLUXFo) - [3.3 Sine and Cosine Function Values](/ap-pre-calc/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) - [3.4 Sine and Cosine Function Graphs](/ap-pre-calc/unit-3/sine-cosine-function-graphs/study-guide/z43MPKoTrSsrfq2Re2Ws) - [3.5 Sinusoidal Functions](/ap-pre-calc/unit-3/sinusoidal-functions/study-guide/lMqyfU03HpgMnHJMRBw4) - [3.6 Sinusoidal Function Transformations](/ap-pre-calc/unit-3/sinusoidal-function-transformations/study-guide/1xRAbpsfqkTOU10kPQ4p) - [3.7 Sinusoidal Function Context and Data Modeling](/ap-pre-calc/unit-3/sinusoidal-function-context-data-modeling/study-guide/NfgWcSvLUIRp9XqiYfQy) - [3.8 The Tangent Function](/ap-pre-calc/unit-3/tangent-function/study-guide/MmhWdpovNDRCpyBgCc0x) - [3.9 Inverse Trigonometric Functions](/ap-pre-calc/unit-3/inverse-trigonometric-functions/study-guide/y9F3Wve0ZJEuOeKJvpP3) - [3.10 Trigonometric Equations and Inequalities](/ap-pre-calc/unit-3/trigonometric-equations-inequalities/study-guide/CAlezrVbYlsGW69J1KcW) - [3.11 The Secant, Cosecant, and Cotangent Functions](/ap-pre-calc/unit-3/secant-cosecant-cotangent-functions/study-guide/nhIcN0Whx8hECmPVKlvK) - [3.12 Equivalent Representations of Trigonometric Functions](/ap-pre-calc/unit-3/equivalent-representations-trigonometric-functions/study-guide/ElEOcRdfZByN7kekt68Z) - [3.13 Trigonometry and Polar Coordinates](/ap-pre-calc/unit-3/trigonometry-polar-coordinates/study-guide/vrD8KOuadisEAqeZVaQS) - [3.14 Polar Function Graphs](/ap-pre-calc/unit-3/polar-function-graphs/study-guide/4Con24QzKXI6SrwldHmX) - [3.15 Rates of Change in Polar Functions](/ap-pre-calc/unit-3/rates-change-polar-functions/study-guide/7CELC0g92mkEmoGw) ## Other Unit Tests - [Unit 1: Polynomial and Rational Functions](/ap-pre-calc/unit-test/1) - [Unit 2: Exponential and Logarithmic Functions](/ap-pre-calc/unit-test/2) ## About This Document Canonical Fiveable pages are available as Markdown at the same path plus `.md`. - [llms.txt](https://fiveable.me/llms.txt): index of Fiveable's sections and URL patterns - [llms-full.txt](https://fiveable.me/llms-full.txt): complete subject and unit listing - [MCP server](https://fiveable.me/mcp): call Fiveable as tools instead of fetching pages (`https://fiveable.me/api/mcp`) - [MCP server for AP teachers](https://fiveable.me/mcp/teachers): a teacher's classes, assignments and AP-rubric grading (`https://fiveable.me/api/mcp/teacher`)