Determining Growth Factor Of Exponential Function F(x) = (1/5)(15^x)

This article delves into the intricacies of exponential functions, focusing on identifying and understanding the growth factor. We'll use the specific example of the function $f(x) = \frac{1}{5}(15^x)$ to illustrate the concept and guide you through the process of determining the growth factor. Understanding exponential functions and their growth factors is crucial in various fields, including finance, biology, and computer science. Let's embark on this journey to unravel the mysteries of exponential growth.

Understanding Exponential Functions

At its core, an exponential function is a mathematical function where the independent variable (often denoted as x) appears as an exponent. The general form of an exponential function is given by:

$f(x) = a * b^x$

where:

• f(x) represents the output of the function for a given input x.
• a is the initial value or the y-intercept of the function (the value of the function when x = 0).
• b is the base of the exponential function, also known as the growth factor (if b > 1) or decay factor (if 0 < b < 1).
• x is the independent variable, representing the exponent.

The growth factor, denoted by b, plays a pivotal role in determining the behavior of the exponential function. When the growth factor is greater than 1 (b > 1), the function exhibits exponential growth, meaning the output f(x) increases rapidly as x increases. Conversely, when the growth factor is between 0 and 1 (0 < b < 1), the function exhibits exponential decay, where the output f(x) decreases as x increases.

To truly grasp the essence of exponential functions, it's essential to understand the interplay between the initial value (a) and the growth factor (b). The initial value sets the starting point of the function, while the growth factor dictates the rate at which the function increases or decreases. A larger growth factor implies a faster rate of increase, while a growth factor closer to 1 indicates a slower rate of change. The initial value a scales the exponential term b^x, effectively stretching or compressing the function vertically. A larger a will result in a larger f(x) for the same x, while a smaller a will result in a smaller f(x). Understanding these parameters allows us to accurately model various real-world phenomena, such as population growth, compound interest, and radioactive decay.

Identifying the Growth Factor

To identify the growth factor in a given exponential function, we need to express the function in the standard form: $f(x) = a * b^x$. Once the function is in this form, the growth factor is simply the value of b. Let's apply this to our example function:

$f(x) = \frac{1}{5}(15^x)$

In this function, we can directly identify the components:

• a = $\frac{1}{5}$ (the initial value)
• b = 15 (the base of the exponent)

Therefore, the growth factor of the function $f(x) = \frac{1}{5}(15^x)$ is 15. This indicates that the function exhibits exponential growth, and for every unit increase in x, the function value is multiplied by 15. This rapid increase is characteristic of exponential growth, making it a powerful tool for modeling phenomena that exhibit rapid change. For instance, consider a population of bacteria that doubles every hour. The growth factor in this scenario would be 2, reflecting the doubling effect. Similarly, in finance, compound interest exhibits exponential growth, where the growth factor is determined by the interest rate. The higher the interest rate, the larger the growth factor, and the faster the investment grows.

Applying the Concept

Now that we've identified the growth factor, let's delve into its significance. The growth factor of 15 in the function $f(x) = \frac{1}{5}(15^x)$ signifies that for every unit increase in x, the function value is multiplied by 15. This highlights the rapid growth characteristic of this exponential function. The initial value, $\frac{1}{5}$, determines the starting point of the function's growth. Understanding the growth factor allows us to predict the function's behavior and values for different inputs. For instance, we can estimate how the function's output will change as x increases or decreases. A large growth factor, like 15, indicates a steep upward curve on the graph of the function, signifying a rapid increase in the output values. Conversely, a smaller growth factor (greater than 1) would result in a less steep curve, indicating a slower rate of growth.

Consider the function's value at x = 0:

$f(0) = \frac{1}{5}(15^0) = \frac{1}{5}(1) = \frac{1}{5}$

This confirms that the initial value, $\frac{1}{5}$, is the function's value when x is 0. Now, let's examine the function's value at x = 1:

$f(1) = \frac{1}{5}(15^1) = \frac{1}{5}(15) = 3$

Notice that the function value has increased from $\frac{1}{5}$ to 3, which is a multiplication by 15 (the growth factor). This illustrates how the growth factor dictates the rate of change in the function's output. By understanding the growth factor, we can gain valuable insights into the function's behavior and its potential applications in various scenarios. For example, in financial modeling, a growth factor can represent the rate of return on an investment, while in population dynamics, it can represent the rate of population increase.

Choosing the Correct Answer

Based on our analysis, the growth factor of the function $f(x) = \frac{1}{5}(15^x)$ is 15. Therefore, the correct answer is:

D. 15

This confirms our understanding of exponential functions and the process of identifying the growth factor. By recognizing the standard form of an exponential function and isolating the base of the exponent, we can easily determine the growth factor and gain valuable insights into the function's behavior. This skill is crucial for analyzing and interpreting exponential models in various fields, from finance and biology to computer science and engineering. The ability to identify the growth factor empowers us to make informed predictions and decisions based on exponential trends.

Conclusion

In conclusion, understanding the growth factor is essential for working with exponential functions. By recognizing the standard form of the function and identifying the base of the exponent, we can easily determine the growth factor and interpret its significance. In the case of the function $f(x) = \frac{1}{5}(15^x)$, the growth factor is 15, indicating a rapid exponential growth. Mastering this concept allows us to analyze and apply exponential functions in various real-world scenarios, making informed decisions and predictions based on exponential trends. The growth factor serves as a key indicator of the rate of change in exponential models, enabling us to understand and interpret the behavior of these functions effectively. This knowledge is invaluable in diverse fields, from financial forecasting to scientific modeling, highlighting the importance of a solid grasp of exponential functions and their growth factors.

This exploration underscores the fundamental role of exponential functions in modeling real-world phenomena. From population growth and compound interest to radioactive decay and the spread of information, exponential models provide a powerful framework for understanding and predicting change. The growth factor, as a key parameter in these models, encapsulates the essence of exponential growth or decay, allowing us to quantify the rate of change and make informed projections. By mastering the concepts and techniques discussed in this article, you will be well-equipped to tackle a wide range of problems involving exponential functions and their applications.