Introduction

3.5 Sinusoidal functions is an essential topic in precalculus and mathematics in general. They model periodic phenomena such as sound waves or tides. The sine and cosine functions are the primary examples of sinusoidal functions, showing the smooth, repetitive oscillation that defines this category. As you prepare for the AP® Precalculus exam, understanding the sinusoidal function and its characteristics will be crucial.

What is a Sinusoidal Function?

A sinusoidal function is any function that involves additive and multiplicative transformations of the sine function. Even cosine is technically a sinusoidal function because it can be expressed as a phase shift of sine.

• Additive Transformations: These shift the graph vertically, affecting the midline.
• Multiplicative Transformations: These alter the amplitude and period of a function by stretching or compressing the graph.

Key Characteristics of Sinusoidal Functions

Amplitude

The amplitude is the height of the wave from the midline to its peak. It measures the function’s intensity or strength.

• Calculation: The amplitude is half the difference between the maximum and minimum values of the function.
• Example: Consider the function y = 3 \sin(x). The amplitude here is simply 3 since it’s the coefficient of the sine.
• Practice Problem: Find the amplitude of y = 5 \cos(x) + 2. Solution: The amplitude is 5.

Period and Frequency Relationship

The period of a sinusoidal function is the length of one complete cycle, while frequency is the number of cycles in a given length.

• Relationship: For sine and cosine, the period is 2\pi, and the frequency is 1.
• Example: Find the period of y = \sin(2x). Solution: The period is 2\pi/2 = \pi.
• Practice Problem: Determine the period and frequency of y = \cos(\frac{1}{3}x). Solution: Period = 2\pi \times 3 = 6\pi, Frequency = 1/3.

Midline

The midline of a sinusoidal function is the horizontal line that runs equidistant from the peaks and troughs.

• How to Find the Mideline of a Trig Function: It’s the average of the maximum and minimum values of the function.
• Example: For y = \sin(x) + 4, the midline is y = 4.
• Practice Problem: Calculate the midline of y = 2 \cos(x) - 1. Solution: The midline is y = -1.

Symmetry

Sinusoidal functions exhibit symmetry:

• Rotational Symmetry: Functions like \sin(x) display rotational symmetry at the origin.
• Reflective Symmetry: On the other hand, \cos(x) displays reflexive symmetry over the y-axis.
• Is cos even or odd? Is sin even or odd?: Sine is odd, meaning \sin(-x) = -\sin(x), while cosine is even, so \cos(-x) = \cos(x).

Behavior of Graphs

Sinusoidal graphs oscillate between concave up and concave down, producing distinct waves.

• Visual Representation: These graphs look like hills and valleys that repeat periodically.
• Concavity: Because of this oscillation, the graph continuously changes between concave up and concave down over the intervals.
• Example: Consider f(x)=-2\sin(x) . Just on the interval from [0,2 \pi] , the graph oscillates from concave up on [0, \pi] and concave down on [\pi,2 \pi] . This pattern continues infinitely.

Quick Reference Chart: Vocabulary and Definitions

Example Problems and Solutions

• Example 1: Determine the amplitude, period, and midline for y = 3\sin(4x) + 1. Solution: Amplitude = 3, Period = 2\pi/4 = \pi/2, Midline = y = 1.
• Example 2: Graph the function y = 2\cos(x + \pi) - 3. Solution: Amplitude = 2, Period = 2\pi, Shifted left by \pi, Midline = y = -3.

Conclusion

This guide covers the essential characteristics that make sinusoidal functions vital in understanding periodic phenomena. Mastering concepts like amplitude, period, and symmetry helps interpret situations modeled by these functions. Practicing these skills will enhance understanding and success in calculus and beyond.

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