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3.6 Sinusoidal Function Transformations

4 min read•june 18, 2024

Kashvi Panjolia

We know that all -- functions that look like the sine curve -- have certain common characteristics. We qualitatively defined amplitude, period, , , and other terms in the last guide, and now we are going to define them quantitatively. We will learn new terms as well, and focus on how they all connect through the . By the end of this guide, you will know this equation inside and out, and be able to get loads of information from it.

The Equation

The equation for a sinusoidal function that is based on the sine curve is

Image courtesy of CollegeBoard.

The equation for a sinusoidal function that is based on the cosine curve is

Image courtesy of CollegeBoard.

These equations may look scary at first, but they are really very similar to what you learned in algebra. The main purpose of all the letters in the equation is to show the transformations of the sinusoidal function from the original sine (or cosine) curve. These transformations are , , , and . All the transformations also apply to the cosine curve in the same way, but we will be using sine in this guide for simplicity. Let's go through the equations above one part at a time to discover the plethora of information they contain.

The equation f(θ) = asin(b(θ + c)) + d is made up of four parts: the function f(θ), the variable θ, the , and the constants a, b, c, and d. The function f(θ) is the overall function that we are trying to solve. The variable θ is the input to the function, and the output is the value of f(θ). The variable x can also be used to represent the angle in this equation, and the output would become the value of f(x).

The trigonometric function sin, which stands for "sine," is a mathematical function that takes an angle and returns the sine of that angle. In this equation, the sin function is applied to the expression b(θ + c), which is inside the parentheses.

The constants a, b, c, and d are used to manipulate the equation to fit a specific situation. These constants can be used to change different characteristics of the wave pattern that is being represented by the equation.

Amplitude

The constant "a" represents the amplitude of the wave. The amplitude is the measure of how high the wave is from its resting position, or midline. It is the maximum displacement of a point on the wave from its undisturbed position. When the value of "a" is increased, the amplitude of the wave also increases, making the wave taller. When the value of "a" is decreased, the amplitude of the wave decreases, making the wave shorter.

The "a" value in the equation may be negative, but the amplitude is always positive. If the value for "a" were -7, you would say the amplitude is 7. The negative sign represents a reflection over the x-axis of the wave.

Period

The constant "b" represents the frequency of the wave, or the wavelength of the wave. The frequency of a wave is the number of oscillations (or cycles) that occur in one second, and it is measured in hertz (Hz). The reciprocal of the frequency is the period of the wave, and it is the amount of time it takes for one complete cycle of the wave to occur.

In the case of a sinusoidal function, the period is given by the equation T = 2𝛑/b where T is the period and b is the frequency (or wavelength). This equation states that the period of the sinusoidal function is equal to twice 𝛑 (2𝛑) divided by the frequency (b).

As the frequency (b) increases, the period of the wave decreases, meaning that the wave oscillates more slowly. Conversely, as the frequency (b) decreases, the period of the wave increases, meaning that the wave oscillates more quickly.

Image courtesy of MathIsFun.

Phase Shift

The constant "c" represents the , or horizontal shift, of the wave. The phase shift is the amount by which the wave is shifted to the left or right. It is the amount of horizontal displacement of the wave. When the value of "c" is increased, the wave is shifted to the right. When the value of "c" is decreased, the wave is shifted to the left.

Vertical Translation

The constant "d" represents the vertical shift of the wave. The vertical shift is the amount by which the wave is shifted up or down. It is the amount of vertical displacement of the wave. When the value of "d" is increased, the wave is shifted upward. When the value of "d" is decreased, the wave is shifted downward.

Below is an image showing how the equation for a sinusoidal function can be constructed from a graph:

Image courtesy of MathIsFun.

Practice Problems

1. What is the amplitude of the wave represented by the equation f(θ) = 3sin(2(θ+1)) + 5? a) 3

b) 2

c) 5

d) 1

Answer: a) 3

2. What is the wavelength of the wave represented by the equation f(θ) = 2sin(0.5(θ-2)) + 3? a) 2π

b) 4π

c) 0.5π

d) 1π

Answer: c) 0.5π

3. What is the phase shift of the wave represented by the equation f(θ) = 4sin(3θ+1.5)) - 2? a) 0.5

b) -2

c) 3

d) 1.5

Answer: a) 0.5

Note: For this problem, there is a 3 in front of the θ inside the parentheses. To find the "c" value, or the phase shift, we need to factor out the 3 so that we get θ by itself. By factoring out the 3 from 1.5, we found the phase shift to be 0.5.

Key Terms to Review (11)

Equation for a Sinusoidal Function: The equation for a sinusoidal function represents the mathematical form of sine and cosine waves, commonly expressed as $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$. These equations incorporate various transformations that affect the amplitude, period, phase shift, and vertical shift of the wave, allowing for the representation of a wide range of real-world periodic phenomena.

Frequency: Frequency refers to the number of complete cycles of a periodic function that occur in a unit of time. In the context of sinusoidal functions, frequency determines how often the function repeats itself over a specified interval, influencing its overall shape and behavior. Understanding frequency is essential for analyzing oscillatory phenomena and modeling real-world data, as it connects directly to amplitude, period, and phase shifts within these functions.

Horizontal Shift: A horizontal shift refers to the movement of a function along the x-axis, either to the left or right, without changing its shape or orientation. This transformation is crucial for understanding how periodic functions, like sine, cosine, and tangent functions, can be adjusted to model real-world phenomena by altering their starting points. The horizontal shift is determined by the value added to or subtracted from the input variable of the function, affecting where the function begins its cycle.

Horizontal Stretch/Compression: Horizontal stretch and compression refer to the transformations that affect the width of a function's graph, specifically sinusoidal functions like sine and cosine. A horizontal stretch occurs when the graph is stretched away from the y-axis, making it wider, while a horizontal compression occurs when the graph is compressed toward the y-axis, making it narrower. These transformations are essential for understanding how sinusoidal functions can be manipulated to fit different scenarios or data.

Midline: The midline is a horizontal line that represents the average value or center position of a sinusoidal function, effectively dividing the graph into two equal halves. It serves as a reference point for the amplitude, which is the distance from the midline to either the maximum or minimum point of the wave. Understanding the midline is crucial for analyzing transformations in sinusoidal functions, including vertical shifts and amplitude adjustments.

Phase Shift: Phase shift refers to the horizontal translation of a periodic function, indicating how far the function has been shifted from its original position along the x-axis. This concept is essential in understanding how functions like sine, cosine, and tangent can be adjusted to fit various contexts, such as modeling real-world data or analyzing transformations. The phase shift can be positive or negative, affecting the starting point of the wave and altering the timing of the peaks and troughs.

Sinusoidal Functions: Sinusoidal functions are periodic functions that describe smooth, wave-like patterns and are represented mathematically by sine and cosine functions. These functions are characterized by their amplitude, period, phase shift, and vertical shift, making them essential for modeling real-world phenomena such as sound waves, light waves, and seasonal changes. Their transformations allow for modifications of these attributes to fit various applications.

Trigonometric Function Sine: The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. It plays a crucial role in understanding periodic behavior and is essential when analyzing sinusoidal functions and their transformations, such as amplitude, phase shift, and vertical shifts. The sine function has a range of values from -1 to 1, creating smooth, wave-like graphs that repeat at regular intervals.

Vertical Shift: A vertical shift refers to the upward or downward movement of a function's graph along the y-axis, resulting from the addition or subtraction of a constant value to the function's output. This transformation affects the overall position of the graph but does not change its shape or periodic characteristics. By applying a vertical shift, one can modify the baseline of the function, which is essential in analyzing data that may require adjustments for more accurate modeling.

Vertical Stretch/Compression: Vertical stretch and compression refer to the transformation of a function's graph by multiplying its output values by a constant factor. When a function is vertically stretched, its graph becomes taller and the peaks and valleys of the function increase in distance from the horizontal axis. Conversely, vertical compression makes the graph flatter, decreasing the distance of these points from the horizontal axis. Understanding these transformations helps in analyzing sinusoidal functions and their behaviors.

Vertical Translation: Vertical translation refers to the process of shifting a function up or down along the y-axis without changing its shape. This transformation is represented mathematically by adding or subtracting a constant value to the function's output, effectively adjusting its baseline. Understanding vertical translation is essential as it allows for the manipulation of sinusoidal functions, affecting their range and midline position while preserving other characteristics such as amplitude and period.

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