AP Precalculus Practice Test: Unit 3 Question #2 Location of a Later Value on a Periodic Graph

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

**AP Precalculus Practice Test: Unit 3 Question #2** often involves determining the location of a later value on a periodic graph, such as finding the height, position, or other attributes of a point on a sinusoidal or trigonometric function after a specified interval. Here's an example and the relevant concepts:

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### **Problem Description**

The problem typically describes a situation modeled by a periodic function. For example:

*A buoy in the ocean moves up and down in a sinusoidal pattern. Its height above sea level is modeled by the function \( h(t) = 2 \sin(\pi t) + 5 \), where \( h(t) \) is the height in meters at time \( t \) in seconds. Find the height of the buoy at \( t = 2.5 \) seconds.*

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### **Key Concepts and Vocabulary**

1. **Periodic Function**: A function that repeats its values at regular intervals, such as sine or cosine. Important terms include:
- **Amplitude**: Maximum displacement from the midline (\(2\) meters in this case).
- **Midline**: The average height (\(5\) meters).
- **Period**: The time for one complete cycle (\( \text{Period} = \frac{2\pi}{\pi} = 2 \) seconds).

2. **Evaluating the Function**: To find the height at \( t = 2.5 \), substitute \( t = 2.5 \) into the given equation:
\[
h(2.5) = 2 \sin(\pi \cdot 2.5) + 5
\]

3. **Reference Angles and Cycles**: For periodic functions, angles may exceed \( 2\pi \), so understanding how values repeat and relate to reference angles is essential.

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### **Solution to Example**

Substitute \( t = 2.5 \) into \( h(t) \):
\[
h(2.5) = 2 \sin(\pi \cdot 2.5) + 5
\]
\[
h(2.5) = 2 \sin(2.5\pi) + 5
\]
Since \( 2.5\pi \) corresponds to \( 1.5\pi \) after removing complete cycles (\( 2\pi \)), the sine value at \( 1.5\pi \) is \( -1 \):
\[
h(2.5) = 2(-1) + 5 = -2 + 5 = 3
\]

The height of the buoy at \( t = 2.5 \) seconds is **3 meters**.

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### **Purpose of the Question**
This type of question assesses your ability to:
- Interpret the properties of a periodic graph.
- Apply the sine or cosine function to specific times or positions.
- Use transformations (amplitude, midline, and phase shifts) and the function’s periodic nature to solve problems.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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