Best Description Of The Graph Of F(x) = 60(1/3)^x

#title: Graphing Exponential Functions A Detailed Analysis of f(x) = 60(1/3)^x

In this article, we delve deep into the analysis of the function f(x) = 60(1/3)^x, exploring its graphical representation and key characteristics. This function is a classic example of exponential decay, and understanding its behavior is crucial in various fields, from mathematics and physics to finance and computer science. Our primary focus will be to identify the best description of the graph, dissecting its initial value, rate of decay, and overall trend. We will meticulously examine each component of the function, highlighting how they contribute to the graph's unique shape and properties. This comprehensive analysis aims to provide a clear and insightful understanding of exponential decay functions and their graphical representations.

Dissecting the Exponential Function: f(x) = 60(1/3)^x

To understand the graph of the function f(x) = 60(1/3)^x, we must first dissect its components. This function is an exponential function, characterized by a constant base raised to a variable exponent. In this case, the function can be broken down into two primary components:

1. Initial Value: The coefficient 60 represents the initial value of the function. This is the value of f(x) when x is equal to 0. In other words, it's the y-intercept of the graph. The initial value plays a critical role in determining the starting point of the graph and significantly influences its overall scale.

2. Decay Factor: The term (1/3)^x is the exponential part of the function. The base, 1/3, is a fraction between 0 and 1, which indicates that the function represents exponential decay. As x increases, the value of (1/3)^x decreases, causing the function f(x) to decrease as well. This decay factor is crucial in determining the rate at which the function decreases. A smaller base (closer to 0) results in a faster rate of decay, while a larger base (closer to 1) results in a slower rate of decay.

The initial value of 60 signifies that the graph starts at the point (0, 60) on the Cartesian plane. This point is the highest point on the graph, as the function decreases as x increases. The decay factor of 1/3 indicates that for each unit increase in x, the function's value is multiplied by 1/3. This means that the function decreases rapidly at first, then slows down as it approaches the x-axis. The graph will never actually touch the x-axis, as the function will always have a positive value, no matter how large x becomes. This behavior is characteristic of exponential decay functions.

Understanding these components is essential for accurately describing and interpreting the graph of the function. The initial value sets the stage, while the decay factor dictates the function's long-term behavior. Together, they paint a clear picture of a function that starts at 60 and decreases exponentially towards zero.

Identifying the Correct Graph Description

Given the function f(x) = 60(1/3)^x, let's analyze possible descriptions of its graph to pinpoint the most accurate one. We need to focus on the initial value and the nature of the exponential change.

Option A suggests an initial value of 20 and a constant subtraction of 1/3. This description is incorrect. The initial value of the function f(x) = 60(1/3)^x is determined by substituting x = 0 into the equation, which yields f(0) = 60(1/3)^0 = 60 * 1 = 60. Therefore, the initial value is 60, not 20. Furthermore, the function does not decrease by a constant amount of 1/3. Instead, it decreases by a factor of 1/3 at each step, which is a multiplicative process, characteristic of exponential functions, not a linear one involving subtraction.

Another common misconception is to confuse the decay factor with a constant rate of decrease. The function's value is multiplied by 1/3 for each unit increase in x, resulting in a decreasing value, but this is not a simple subtraction. The rate of decrease is proportional to the current value of the function, leading to a curve that flattens out as x increases.

The correct description must highlight the initial value of 60 and the multiplicative decay factor of 1/3. It should accurately convey that the function's value is being repeatedly multiplied by 1/3, leading to exponential decay. This understanding is crucial for interpreting the graph's behavior and making accurate predictions about the function's values at different points.

Key Characteristics of the Graph: Exponential Decay

The graph of f(x) = 60(1/3)^x exhibits several key characteristics that are typical of exponential decay functions. Understanding these characteristics is crucial for interpreting the function's behavior and making predictions about its values.

1. Initial Value: As previously discussed, the graph starts at the point (0, 60). This is the y-intercept and the highest point on the graph, representing the function's value when x is zero. The initial value provides a crucial reference point for understanding the function's overall scale.

2. Exponential Decay: The graph decreases as x increases, demonstrating exponential decay. This decay is characterized by a rapid decrease at first, followed by a gradual flattening out as the function approaches the x-axis. The function never actually touches the x-axis, as it will always have a positive value, no matter how large x becomes. This asymptotic behavior is a hallmark of exponential decay functions.

3. Rate of Decay: The rate of decay is determined by the base of the exponential term, which is 1/3 in this case. For each unit increase in x, the function's value is multiplied by 1/3. This means that the function decreases by a factor of 1/3 at each step. The rate of decay is constant in the sense that the multiplicative factor is constant, but the amount of decrease changes with x. For example, the decrease from x = 0 to x = 1 is significantly larger than the decrease from x = 10 to x = 11, demonstrating the decreasing rate of decay.

4. Asymptotic Behavior: The graph approaches the x-axis (y = 0) as x approaches infinity. This means that the function's value gets closer and closer to zero but never actually reaches it. This asymptotic behavior is a key characteristic of exponential decay functions and distinguishes them from linear or polynomial functions.

These key characteristics collectively paint a clear picture of the function's behavior. The graph starts at 60, decreases exponentially towards zero, and exhibits a decreasing rate of decay as x increases. This understanding is essential for applying exponential decay functions to real-world scenarios and interpreting their results.

Real-World Applications of Exponential Decay

Exponential decay functions, like f(x) = 60(1/3)^x, have numerous applications in various fields. Understanding these applications provides context and highlights the practical significance of exponential decay.

1. Radioactive Decay: Radioactive substances decay exponentially, meaning that the amount of substance decreases over time according to an exponential function. The half-life of a radioactive substance is the time it takes for half of the substance to decay. This concept is widely used in nuclear physics, geology, and archaeology for dating materials.

2. Drug Metabolism: The concentration of a drug in the bloodstream typically decreases exponentially over time as the body metabolizes and eliminates the drug. This is a crucial consideration in pharmacology and medicine for determining drug dosages and dosing intervals.

3. Population Decline: In some cases, populations can decline exponentially due to factors such as disease, habitat loss, or overhunting. Understanding exponential decay can help scientists and conservationists model and manage population decline in endangered species.

4. Financial Applications: Exponential decay can be used to model the depreciation of assets or the decay of investment value due to inflation or other factors. This is an important concept in finance and accounting.

5. Electrical Circuits: In electrical circuits, the charge on a capacitor or the current in an inductor can decay exponentially over time. This is a fundamental concept in electrical engineering.

These examples illustrate the widespread applicability of exponential decay functions. By understanding the mathematical principles behind exponential decay, we can better model and analyze real-world phenomena in various fields.

Conclusion: The Significance of Initial Value and Decay Factor

In conclusion, the function f(x) = 60(1/3)^x provides a compelling example of exponential decay. The best description of its graph accurately reflects the initial value of 60 and the decay factor of 1/3. This means the graph starts at the point (0, 60) and decreases exponentially as x increases, with its value being multiplied by 1/3 for each unit increase in x.

We have explored the key characteristics of the graph, including its exponential decay, rate of decay, and asymptotic behavior. We have also examined the real-world applications of exponential decay, highlighting its relevance in fields such as radioactive decay, drug metabolism, population decline, finance, and electrical circuits.

The initial value and decay factor are the two most important parameters that define an exponential decay function. The initial value determines the starting point of the graph, while the decay factor dictates the rate at which the function decreases. Understanding these parameters is crucial for interpreting the function's behavior and making predictions about its values.

By thoroughly analyzing the function f(x) = 60(1/3)^x, we have gained a deeper understanding of exponential decay and its graphical representation. This knowledge is valuable in various contexts, from solving mathematical problems to modeling real-world phenomena. The ability to identify and interpret exponential decay functions is an essential skill in mathematics, science, and engineering.