AP Precalculus Test Prep - The Period of a Sinusoidal Function (Multiple Choice)

To visually find the period of a sinusoidal function by looking at its graph, follow these steps:

1. **Identify the Function:** Determine if the graph represents a sine or cosine function. Sine functions start at the origin, while cosine functions start at their maximum or minimum.

2. **Identify One Complete Cycle:** Locate one complete cycle of the sinusoidal function on the graph. This includes a full wave from a peak (or trough) to the next peak (or trough).

3. **Measure the Horizontal Distance:** Measure the horizontal distance between two consecutive peaks (or troughs) that form a complete cycle. This distance is the period of the sinusoidal function.

4. **Amplitude Effects:** Keep in mind that the amplitude of the function (the vertical distance between the midline and a peak or trough) doesn't affect the period. Only the horizontal spacing between cycles determines the period.

5. **Shifts and Reflections:** If the graph is shifted horizontally (phase shift) or reflected across the x-axis (negation of amplitude), these transformations do not affect the period either. The period remains the same despite these alterations.

6. **Fractional Periods:** If the graph contains a fraction of a complete cycle, ensure that you're measuring the distance between two consecutive points that make up a full wave.

7. **Axis Labels:** Pay attention to the labeling of the x-axis on the graph. It should provide numerical information that matches the period you've visually determined.

8. **Check Your Answer:** To verify your visual determination, you can calculate the period using the formula: period = \( \frac{2\pi}{\text{angular frequency}} \), where the angular frequency is related to the coefficient of \(x\) in the function.

Remember that the period of a sinusoidal function is the distance it takes for the function to repeat its pattern. By identifying and measuring one complete cycle on the graph, you can accurately determine the period of the function visually.
These videos are examples taken from AP Precalculus Test study guides or official course description guides created by Collegeboard.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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