Problem 3 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

What is a phase shift? Give an example of a sine function that has a phase shift.

Step-by-Step Solution

Answer

A phase shift refers to the horizontal shift that occurs when a trigonometric function is translated left or right. An example of a sine function with a phase shift is \(y = \sin(x - \pi /2)\), which has a phase shift of \(\pi /2\), shifting the function to the right by \(\pi /2\) units.

Step 1: Definition of Phase Shift

Phase shift refers to the horizontal shift of a wave, in this case, a trigonometric function, either sine or cosine, that occurs when the function is translated left or right. This shifts the starting point of the periodic function.

Step 2: Basic Form of a Sine Function

The standard form of a sine function is \(y = A \sin(B(x - C)) + D\). In this equation, A denotes the amplitude, B affects the period, C is the phase shift, and D is the vertical shift.

Step 3: Example and Calculation

An example of a sine function with a phase shift could be \(y = \sin(x - \pi /2)\). In this case, the phase shift is \(\pi /2\), which means that the graph of the sine function will be shifted to the right by \(\pi /2\) units compared to the basic sine function \(y = \sin(x)\). So instead of beginning the cycle at \(x = 0\), it would start at \(x = \pi /2\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is represented mathematically as \(y = \sin(x)\), which describes waves or oscillations. The standard sine wave starts at the origin, moves up to its maximum value at \(\frac{\pi}{2}\), returns to zero, goes to its minimum at \(\frac{3\pi}{2}\), and completes the cycle at \(2\pi\).
It is often used to model periodic phenomena, such as sound waves or tides.
The function repeats its pattern over a fixed interval, which is crucial for understanding waves in various physical contexts.
Understanding the sine function's behavior is essential in studying how shifts and changes in amplitude affect the graph.

Trigonometric Functions

Trigonometric functions include sine, cosine, and tangent, among others. These functions are essential in mathematics, especially in the study of angles and periodic phenomena.
Each function has its unique properties and uses, but all relate to the angles of a right triangle or the unit circle.
For example, in a right triangle:

• The sine function represents the ratio of the opposite side to the hypotenuse.
• Cosine is the adjacent side over the hypotenuse.
• Tangent is the ratio of the opposite side to the adjacent side.

Beyond triangles, these functions help describe waves and oscillations. Understanding them is vital for analyzing any periodic phenomena.

Horizontal Shift

A horizontal shift in a trigonometric function, often referred to as a phase shift, translates the graph left or right.
In the function \(y = A \sin(B(x - C)) + D\), the term \(C\) represents this shift.
When \(C > 0\), the graph shifts to the right. If \(C < 0\), it shifts to the left. This adjustment modifies where the wave begins its cycle.
_Example:_

• Consider the function \(y = \sin(x - \pi/2)\). The graph is shifted to the right by \(\pi/2\) units compared to \(y = \sin(x)\).
  
• Contrast this with \(y = \sin(x + \pi/2)\), where the graph shifts \(\pi/2\) units to the left.

Mastering how to handle phase shifts is crucial for precisely tuning models to match real-world data.

Periodic Function

Periodic functions are mathematical functions that repeat their values in regular intervals, or periods.
A well-known example is the sine function, \(y = \sin(x)\), which has a period of \(2\pi\).
This means every \(2\pi\) units, the function's values repeat.

• **Period:** The length of one complete cycle of the function.
• **Amplitude:** The height from the centerline to the peak or trough of the wave.
  

Alternating functions, like sine and cosine, are perfect examples of periodic behavior, essential in fields such as physics and engineering.
They provide predictability and consistency, vital for modeling cyclical processes like seasonal temperatures or sound vibrations.