Sketching Y = |2cos(3/2 X) - 1| Graph For 0 ≤ X ≤ 2π

Introduction

In this article, we will delve into the process of sketching the graph of the function y = |2cos(3/2 x) - 1| within the interval 0 ≤ x ≤ 2π. This involves understanding the transformations applied to the basic cosine function and the effect of the absolute value. We will break down the function step by step, analyzing its key features such as amplitude, period, and phase shift, and how these characteristics influence the final shape of the graph. By carefully considering these elements, we can accurately sketch the graph and gain a deeper understanding of the function's behavior. This exploration will not only enhance our graphing skills but also provide valuable insights into the properties of trigonometric functions and their transformations. Let's embark on this graphical journey and unravel the intricacies of this fascinating function.

Understanding the Base Function: y = cos(x)

Before we tackle the given function, y = |2cos(3/2 x) - 1|, it's crucial to understand the basic cosine function, y = cos(x). The cosine function is a fundamental trigonometric function that oscillates between -1 and 1. Its graph has a characteristic wave-like shape, with a period of 2π, meaning the pattern repeats every 2π units along the x-axis. The amplitude of the cosine function is 1, which represents the distance from the midline (the x-axis in this case) to the maximum or minimum point of the wave. The cosine function starts at its maximum value of 1 at x = 0, decreases to 0 at x = π/2, reaches its minimum value of -1 at x = π, returns to 0 at x = 3π/2, and completes one full cycle at x = 2π. Understanding these key features – the period, amplitude, and key points – of the basic cosine function is essential because they serve as the foundation for understanding transformations applied to it.

To truly grasp the cosine function, let's dive deeper into its properties. The cosine function is an even function, meaning that cos(-x) = cos(x). This symmetry is reflected in its graph, which is symmetrical about the y-axis. This property is useful when sketching or analyzing cosine functions, as it allows us to focus on one half of the graph and then mirror it to the other side. Furthermore, the cosine function is closely related to the sine function; in fact, cos(x) = sin(x + π/2). This relationship highlights the phase shift between the two functions, where the cosine function is essentially a sine function shifted by π/2 units to the left. By keeping these fundamental aspects of the cosine function in mind, we are better equipped to analyze and sketch transformations of it, such as the one presented in the original question.

Analyzing the Transformations: y = 2cos(3/2 x) - 1

Now, let's break down the transformations applied to the basic cosine function in y = 2cos(3/2 x) - 1. There are three key transformations to consider: the vertical stretch by a factor of 2, the horizontal compression by a factor of 3/2, and the vertical shift downwards by 1 unit. The vertical stretch by a factor of 2 affects the amplitude of the cosine function. Instead of oscillating between -1 and 1, the function will now oscillate between -2 and 2. This means the maximum value will be 2, and the minimum value will be -2. The horizontal compression by a factor of 3/2 affects the period of the function. The original period of cos(x) is 2π, but the period of cos(3/2 x) is 2π / (3/2) = 4π/3. This means that the function completes one full cycle in a shorter interval than the basic cosine function, resulting in a compressed graph. The vertical shift downwards by 1 unit shifts the entire graph down by 1 unit. This means the midline of the function, which was originally the x-axis, is now the line y = -1.

Understanding the order in which these transformations are applied is also crucial. Generally, stretches and compressions are applied before shifts. In this case, the vertical stretch and horizontal compression are applied to the basic cosine function first, followed by the vertical shift. This order ensures that the transformations are applied correctly and that the resulting graph accurately reflects the function. By carefully considering each transformation and its effect on the graph, we can begin to visualize the shape of y = 2cos(3/2 x) - 1. The graph will be a cosine wave with a larger amplitude, a shorter period, and a shifted midline. Identifying these key changes allows us to accurately plot the points and sketch the graph, bringing us one step closer to understanding the final function.

Incorporating the Absolute Value: y = |2cos(3/2 x) - 1|

The final step in sketching the graph is to consider the absolute value: y = |2cos(3/2 x) - 1|. The absolute value function, denoted by the vertical bars, takes any input and returns its non-negative value. In the context of graphing, this means that any part of the graph that lies below the x-axis (i.e., where y is negative) will be reflected across the x-axis. The portions of the graph that are already above the x-axis remain unchanged. This transformation significantly alters the shape of the graph, as it eliminates any negative y-values and creates a graph that is entirely above or on the x-axis. The points where the original graph intersects the x-axis remain the same, as the absolute value of 0 is still 0. The minimum values of the original function, which were negative, become the maximum values of the transformed function after reflection.

To visualize the effect of the absolute value, imagine taking the graph of y = 2cos(3/2 x) - 1 and folding it along the x-axis. The parts of the graph below the x-axis will flip upwards, creating a mirror image above the x-axis. This results in a graph that has no negative y-values and appears to have