Period And Range Of G(x) = -4cos(πx/3) - 2 A Comprehensive Analysis

In this comprehensive guide, we will delve into the intricacies of the function g(x) = -4cos(πx/3) - 2. Our primary focus will be on determining two key characteristics of this trigonometric function: its period and its range. These concepts are fundamental to understanding the behavior and graphical representation of trigonometric functions. By the end of this exploration, you will have a solid grasp of how to calculate these properties and what they signify in the context of this specific function.

The period of a trigonometric function is the horizontal distance over which the function's graph completes one full cycle. In simpler terms, it's the length of the interval after which the function's values start repeating. For the cosine function, the standard period is 2π. However, when the argument of the cosine function is multiplied by a constant, the period changes. Understanding how to calculate this new period is crucial for analyzing trigonometric functions.

To find the period of g(x) = -4cos(πx/3) - 2, we need to consider the coefficient of x inside the cosine function. In this case, the coefficient is π/3. The general formula for the period of a cosine function of the form cos(Bx) is given by 2π/|B|. Applying this formula to our function, we get:

Period = 2π / |π/3|

To simplify this expression, we divide 2π by the absolute value of π/3:

Period = 2π / (π/3)

To divide by a fraction, we multiply by its reciprocal:

Period = 2π * (3/π)

Now, we can cancel out the π terms:

Period = 2 * 3

Therefore, the period of the function g(x) = -4cos(πx/3) - 2 is 6. This means that the graph of this function will complete one full cycle over an interval of 6 units on the x-axis. This understanding is crucial for accurately graphing the function and predicting its behavior over different intervals.

The range of a function is the set of all possible output values (y-values) that the function can produce. For trigonometric functions, the range is influenced by the amplitude, vertical shifts, and any reflections. Determining the range of g(x) = -4cos(πx/3) - 2 involves understanding how these transformations affect the standard cosine function's range.

The standard cosine function, cos(x), has a range of [-1, 1]. This means that its output values always fall between -1 and 1, inclusive. However, our function g(x) has undergone several transformations that will affect its range. Let's break down these transformations step-by-step:

1. Vertical Stretch/Compression: The cosine function is multiplied by -4. The absolute value of this factor, |–4| = 4, represents the amplitude of the transformed function. This means the function's vertical stretch is 4 times the original cosine function, so the range becomes [-4, 4]. The negative sign indicates a reflection over the x-axis, which means the function will be inverted.

2. Vertical Shift: The function has a constant term of -2 added to it. This represents a vertical shift downward by 2 units. This shift will affect the entire range by moving it down by 2 units.

Now, let's combine these transformations to determine the range of g(x). Starting with the range of the vertically stretched and reflected cosine function, [-4, 4], we apply the vertical shift of -2:

Lower bound: -4 - 2 = -6

Upper bound: 4 - 2 = 2

Therefore, the range of the function g(x) = -4cos(πx/3) - 2 is [-6, 2]. This means that the output values of this function will always fall between -6 and 2, inclusive. Understanding the range is vital for interpreting the function's behavior and its possible output values.

A visual representation of the function can significantly aid in understanding its period and range. If we were to graph g(x) = -4cos(πx/3) - 2, we would observe a cosine wave that has been stretched vertically by a factor of 4, reflected over the x-axis, and shifted downward by 2 units. The graph would complete one full cycle over an interval of 6 units along the x-axis, confirming our calculated period. The highest point of the graph would be at y = 2, and the lowest point would be at y = -6, visually demonstrating the range of the function.

The concepts of period and range are not just theoretical; they have practical applications in various fields. For instance, in physics, understanding the period of a wave is essential for analyzing oscillatory motion, such as the motion of a pendulum or the vibration of a string. In signal processing, the range of a signal indicates the possible values it can take, which is crucial for designing and analyzing communication systems. In economics, periodic functions can model cyclical phenomena, such as seasonal sales patterns, and understanding their range can help in forecasting and decision-making.

In this comprehensive analysis, we have successfully determined the period and range of the function g(x) = -4cos(πx/3) - 2. We found that the period is 6, indicating the function completes one full cycle over an interval of 6 units. We also determined that the range is [-6, 2], representing the set of all possible output values of the function. These properties provide valuable insights into the behavior and characteristics of this trigonometric function.

Understanding the period and range is fundamental to comprehending trigonometric functions and their applications. By mastering these concepts, you can analyze and interpret a wide range of mathematical models and real-world phenomena that exhibit periodic behavior.

We have explored the period and range of the trigonometric function g(x) = -4cos(πx/3) - 2 in detail. This exploration provides a foundational understanding for analyzing other trigonometric functions and their applications. Remember, the period dictates the cyclical nature of the function, while the range defines the boundaries of its output values. By grasping these concepts, you can effectively analyze and interpret trigonometric functions in various contexts.