AP Precalculus Practice Test: Unit 3 Question #26 Describe Translation of a Sinusoidal Function

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 #26** asks you to describe the translation of a sinusoidal function without using inequality symbols. Translating a sinusoidal function involves shifting it horizontally or vertically based on the parameters in its equation, written as \( y = a \sin(b(x - h)) + k \) or \( y = a \cos(b(x - h)) + k \).

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### **Key Components of Translation**
1. **Horizontal Shift (Phase Shift)**:
- Represented by \( h \) in the equation.
- If \( h \) is positive, the graph moves to the right.
- If \( h \) is negative, the graph moves to the left.

2. **Vertical Shift**:
- Represented by \( k \) in the equation.
- If \( k \) is positive, the graph moves upward.
- If \( k \) is negative, the graph moves downward.

3. **No Translation**:
- If \( h \) equals zero, there is no horizontal shift.
- If \( k \) equals zero, there is no vertical shift.

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### **Describing a Translation**
- Analyze the parameters \( h \) and \( k \) in the given sinusoidal equation.
- Use clear descriptive language to explain how the graph is shifted.

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### **Example**
Consider the function \( y = 2 \sin(3(x - 2)) + 4 \):
1. **Horizontal Shift**: The term \((x - 2)\) indicates a shift two units to the right.
2. **Vertical Shift**: The \( + 4 \) indicates a shift four units upward.

Description: "The graph is translated two units to the right and four units upward."

By focusing on the effects of \( h \) and \( k \), you can clearly describe how the graph is transformed without using inequality symbols.

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

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