Finding The Amplitude Of F(x) = (1/2)sin(2x) A Comprehensive Guide

by ADMIN 67 views

In the realm of mathematics, trigonometric functions play a pivotal role, particularly in fields like physics, engineering, and signal processing. Among the fundamental properties of these functions, amplitude stands out as a key characteristic that dictates the vertical extent of their oscillations. This article aims to provide an in-depth understanding of amplitude, focusing on how to determine it from the equation of a trigonometric function, specifically using the example of f(x) = (1/2)sin(2x). We will delve into the definition of amplitude, explore the graphical representation of trigonometric functions, and discuss how transformations affect the amplitude. Furthermore, we will address the given problem by meticulously analyzing the function and identifying its amplitude from the multiple-choice options provided. By the end of this guide, you will have a solid grasp of amplitude and be well-equipped to tackle similar problems involving trigonometric functions.

What is Amplitude?

The amplitude of a trigonometric function, such as sine or cosine, is defined as the distance from the center line (or midline) of the function to the peak (maximum) or trough (minimum) of the wave. It essentially measures the vertical stretch or compression of the function. The center line is the horizontal line that runs midway between the maximum and minimum values of the function. For the basic sine function, f(x) = sin(x), the center line is the x-axis (y = 0), and the amplitude is 1, as the function oscillates between -1 and 1. Similarly, for the basic cosine function, f(x) = cos(x), the amplitude is also 1. Understanding the concept of amplitude is crucial for analyzing and interpreting trigonometric graphs and their corresponding equations. It allows us to quickly grasp the vertical scale of the function and how it deviates from its midline. This is particularly important in applications where trigonometric functions are used to model periodic phenomena, such as sound waves, light waves, and oscillations in mechanical systems. The amplitude directly corresponds to the intensity or strength of these phenomena, making it a critical parameter in their analysis.

Graphical Representation of Trigonometric Functions

To truly grasp the concept of amplitude, it is essential to visualize the graphical representation of trigonometric functions. Let's consider the basic sine function, f(x) = sin(x). Its graph is a wave that oscillates smoothly between -1 and 1. The x-axis represents the angle (usually in radians), and the y-axis represents the value of the sine function at that angle. The wave starts at the origin (0, 0), rises to a maximum value of 1 at x = π/2, returns to 0 at x = π, reaches a minimum value of -1 at x = 3π/2, and completes one full cycle at x = 2π. The amplitude of this function is the distance from the x-axis (the center line) to the highest point (1) or the lowest point (-1), which is 1 unit. Now, let's consider the function given in the problem, f(x) = (1/2)sin(2x). This function is a transformation of the basic sine function. The factor of 1/2 in front of the sine function vertically compresses the graph, while the factor of 2 inside the sine function horizontally compresses it. The vertical compression affects the amplitude, reducing it to 1/2. The horizontal compression affects the period, which is the length of one complete cycle. Visualizing these transformations helps in understanding how they impact the overall shape and characteristics of the trigonometric function. By plotting the graph of f(x) = (1/2)sin(2x), we can clearly see that it oscillates between -1/2 and 1/2, confirming that its amplitude is indeed 1/2.

Transformations and Amplitude

Transformations play a crucial role in altering the amplitude of trigonometric functions. The general form of a transformed sine function is f(x) = A sin(Bx + C) + D, where A, B, C, and D are constants that control various aspects of the function's graph. The constant A directly affects the amplitude. Specifically, the amplitude is given by the absolute value of A, denoted as |A|. If A is greater than 1, the function is vertically stretched, increasing the amplitude. If A is between 0 and 1, the function is vertically compressed, decreasing the amplitude. If A is negative, the function is reflected across the x-axis, but the amplitude remains |A|. The constant B affects the period of the function, which is the length of one complete cycle. The period is given by 2π/|B|. A larger value of |B| results in a shorter period, compressing the graph horizontally. The constant C affects the phase shift, which is the horizontal shift of the graph. The phase shift is given by -C/B. A positive value of -C/B shifts the graph to the left, while a negative value shifts it to the right. The constant D affects the vertical shift, which moves the entire graph up or down. The vertical shift is simply D. Understanding how these transformations interact is essential for accurately interpreting trigonometric functions and their graphs. In the context of the given problem, f(x) = (1/2)sin(2x), we can see that A = 1/2, B = 2, C = 0, and D = 0. Therefore, the amplitude is |1/2| = 1/2, and the period is 2π/2 = π. The function has no phase shift or vertical shift.

Solving the Problem: Determining the Amplitude of f(x) = (1/2)sin(2x)

Now, let's directly address the problem at hand: determining the amplitude of the function f(x) = (1/2)sin(2x). As discussed earlier, the amplitude of a trigonometric function in the form f(x) = A sin(Bx + C) + D is given by the absolute value of A, which is |A|. In this case, A = 1/2, B = 2, C = 0, and D = 0. Therefore, the amplitude is |1/2| = 1/2. Examining the multiple-choice options provided:

  • A. 1/2
  • B. Ï€/4
  • C. Ï€/2
  • D. 1

We can clearly see that option A, 1/2, is the correct answer. The function f(x) = (1/2)sin(2x) oscillates between -1/2 and 1/2, confirming that its amplitude is indeed 1/2. Options B, C, and D are incorrect. π/4 and π/2 are related to the period of the function, not the amplitude, and 1 is the amplitude of the basic sine function, sin(x), but not of the transformed function f(x) = (1/2)sin(2x). This problem highlights the importance of recognizing the coefficient in front of the sine function as the amplitude factor and understanding how it affects the vertical stretch or compression of the graph. By applying this knowledge, we can quickly and accurately determine the amplitude of any trigonometric function in this form.

Conclusion

In conclusion, understanding the amplitude of trigonometric functions is crucial for analyzing their behavior and applications. The amplitude represents the vertical distance from the center line to the maximum or minimum value of the function, and it is directly related to the coefficient in front of the sine or cosine function. Transformations, such as vertical stretches, compressions, and reflections, can alter the amplitude, but the basic principle of identifying the coefficient remains the same. By visualizing the graphs of trigonometric functions and understanding how transformations affect them, we can gain a deeper appreciation for their properties and applications. In the specific case of f(x) = (1/2)sin(2x), the amplitude is 1/2, as determined by the coefficient 1/2 in front of the sine function. This guide has provided a comprehensive overview of amplitude, its definition, graphical representation, and how it is affected by transformations. With this knowledge, you are now well-prepared to tackle a wide range of problems involving trigonometric functions and their amplitudes. Remember to always consider the coefficient in front of the sine or cosine function and apply the concept of |A| to accurately determine the amplitude. Trigonometric functions are fundamental in many areas of mathematics and science, and a strong understanding of their properties, including amplitude, is essential for success in these fields.