Exponential Function H(x) Growth Rate Equation

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In the realm of mathematical functions, exponential functions hold a prominent position, particularly when modeling growth or decay phenomena. Understanding the characteristics of exponential functions and their equations is crucial for various applications, ranging from finance and biology to physics and computer science. This article delves into the specifics of an exponential function, h(x), which exhibits a growth rate of 7.2% and passes through the ordered pair (0, 15). We will explore how to construct the equation that accurately represents this function.

Decoding Exponential Functions

To begin our exploration, let's first understand the general form of an exponential function. An exponential function is typically expressed as:

f(x) = a * b^x

where:

  • f(x) represents the value of the function at a given input x.
  • a denotes the initial value or the y-intercept of the function (the value of the function when x is 0).
  • b is the base of the exponential function, which determines the rate of growth or decay. If b > 1, the function represents exponential growth, and if 0 < b < 1, the function represents exponential decay.
  • x is the independent variable, typically representing time or another relevant quantity.

In the context of our problem, we are given that the exponential function h(x) increases at a rate of 7.2%. This immediately tells us that we are dealing with exponential growth, meaning the base b will be greater than 1. We are also provided with the ordered pair (0, 15), which indicates that when x is 0, the value of the function, h(x), is 15. This ordered pair provides us with the initial value, a, of the function.

Determining the Initial Value (a)

The ordered pair (0, 15) is particularly significant because it directly reveals the initial value of the exponential function. When x = 0, the exponential term b^x becomes b^0, which equals 1. Therefore, h(0) = a * b^0 = a * 1 = a. Since we know that h(0) = 15, we can conclude that the initial value, a, is 15. This means our function will have the form:

h(x) = 15 * b^x

Calculating the Growth Factor (b)

The next step is to determine the growth factor, b. We are given that the function increases at a rate of 7.2%. This percentage increase is crucial for calculating the base of the exponential function. To convert the percentage increase into a decimal, we divide it by 100: 7.2% / 100 = 0.072. This decimal represents the rate of increase.

To find the growth factor b, we add this rate of increase to 1: b = 1 + 0.072 = 1.072. This value represents the factor by which the function's value is multiplied for each unit increase in x. Since b is greater than 1, it confirms that we are indeed dealing with exponential growth. A growth factor of 1.072 indicates that the function's value increases by 7.2% for every unit increase in x.

Constructing the Equation

Now that we have determined both the initial value (a = 15) and the growth factor (b = 1.072), we can construct the equation for the exponential function h(x). Substituting these values into the general form of an exponential function, we get:

h(x) = 15 * (1.072)^x

This equation represents the exponential function that increases at a rate of 7.2% and passes through the ordered pair (0, 15). The equation clearly shows the initial value of 15 and the growth factor of 1.072, which dictates the rate of exponential growth.

Analyzing the Options

Now, let's examine the given options and see which one matches the equation we derived:

A. h(x) = 15(1.928)^x B. h(x) = 15(1.072)^x C. h(x) = 15(0.928)^x D. h(x) = 15(0.072)^x

Comparing these options with our derived equation, h(x) = 15 * (1.072)^x, we can see that option B perfectly matches. The exponential function in option B has the same initial value (15) and the same growth factor (1.072) as our derived equation. Therefore, option B is the correct answer.

Why Other Options Are Incorrect

It's also important to understand why the other options are incorrect:

  • Option A: h(x) = 15(1.928)^x – This option has an incorrect growth factor. The growth factor of 1.928 suggests a much higher growth rate than 7.2%. While the initial value is correct, the exponential term does not reflect the given growth rate.
  • Option C: h(x) = 15(0.928)^x – This option represents exponential decay, not growth. The base of 0.928 is less than 1, indicating that the function's value decreases as x increases. This contradicts the problem statement, which specifies that the function increases at a rate of 7.2%.
  • Option D: h(x) = 15(0.072)^x – This option also represents exponential decay, and the base of 0.072 is significantly less than 1. Additionally, this base does not correctly represent the 7.2% growth rate. It seems to directly use the percentage as the base, which is incorrect.

Practical Applications of the Exponential Function

Understanding exponential functions is not just a mathematical exercise; it has numerous practical applications in various fields. For instance, in finance, exponential functions are used to model compound interest. The growth of an investment over time, where interest is compounded, follows an exponential pattern. In biology, exponential functions can model population growth or the decay of radioactive substances. In physics, exponential functions appear in the study of radioactive decay and the cooling of objects. The applications are vast and varied, highlighting the importance of grasping the concepts of exponential functions.

Key Characteristics of Exponential Functions

To further solidify our understanding, let's summarize the key characteristics of exponential functions:

  • General Form: f(x) = a * b^x
  • Initial Value (a): The value of the function when x = 0. It represents the y-intercept of the function.
  • Base (b): Determines the rate of growth or decay.
    • If b > 1, the function represents exponential growth.
    • If 0 < b < 1, the function represents exponential decay.
  • Growth Factor: For exponential growth, b = 1 + r, where r is the rate of increase (expressed as a decimal).
  • Decay Factor: For exponential decay, b = 1 - r, where r is the rate of decrease (expressed as a decimal).

Conclusion

In summary, we successfully constructed the equation for the exponential function h(x) that increases at a rate of 7.2% and passes through the ordered pair (0, 15). By understanding the general form of exponential functions, determining the initial value and growth factor, and carefully analyzing the given options, we arrived at the correct equation: h(x) = 15(1.072)^x. This exercise underscores the importance of grasping the fundamental principles of exponential functions and their applications in various real-world scenarios. The ability to model and analyze exponential growth and decay is a valuable skill in many disciplines, making the study of these functions a crucial aspect of mathematical education. Through a clear understanding of the components of an exponential function, we can accurately represent and predict phenomena that exhibit exponential behavior. The process of identifying the initial value, calculating the growth or decay factor, and formulating the equation allows us to describe and analyze a wide range of real-world situations, from financial investments to population dynamics and beyond. This understanding provides a powerful tool for making informed decisions and predictions based on mathematical models.

Furthermore, the process of comparing the derived equation with the provided options reinforces the importance of careful analysis and attention to detail. By systematically evaluating each option, we can eliminate incorrect choices and confidently select the one that accurately represents the given conditions. This analytical approach is not only valuable in mathematical problem-solving but also in many other areas of life, where critical thinking and logical reasoning are essential skills.

In conclusion, mastering the concepts of exponential functions and their applications is a valuable investment in one's mathematical and analytical skills. The ability to understand and work with exponential functions opens doors to a wide range of fields and provides a powerful tool for modeling and analyzing the world around us. Whether it's predicting the growth of an investment, understanding population trends, or studying the decay of radioactive materials, exponential functions play a crucial role in our understanding of the world.

By providing a comprehensive explanation of the exponential function h(x), its properties, and its applications, this article aims to enhance the reader's understanding and appreciation of this important mathematical concept. The clear and step-by-step approach, along with the detailed analysis of the given options, ensures that the reader can grasp the key principles and apply them to similar problems in the future. This knowledge will undoubtedly be beneficial in both academic and practical settings, empowering individuals to tackle real-world challenges with confidence and competence.