Finding A And B An Exponential Function Problem F(x) = A * B^x

At the heart of many natural phenomena lies the exponential function, a powerful tool for modeling growth and decay. From population dynamics to compound interest, exponential functions provide a framework for understanding systems that change at a rate proportional to their current value. In this article, we'll delve into the characteristics of exponential functions, explore how to determine their parameters, and work through a specific example to solidify our understanding. We will solve the problem of determining the values of a and b in the exponential function f(x) = a * b^x given that it passes through the points (0, 5) and (2, 20).

Understanding Exponential Functions: The Building Blocks

An exponential function is generally expressed in the form f(x) = a * b^x, where:

• f(x) represents the output value of the function for a given input x.
• a is the initial value or the y-intercept, representing the value of the function when x is 0. It determines the starting point of the exponential curve.
• b is the base or the growth/decay factor. It determines how quickly the function increases or decreases. If b > 1, the function represents exponential growth; if 0 < b < 1, the function represents exponential decay.
• x is the independent variable, often representing time or another quantity that influences the function's value.

The base b is crucial in determining the function's behavior. A base greater than 1 indicates exponential growth, where the function's value increases rapidly as x increases. Examples include population growth and compound interest. Conversely, a base between 0 and 1 indicates exponential decay, where the function's value decreases rapidly as x increases. Examples include radioactive decay and the cooling of an object. The constant a acts as a vertical stretch or compression, scaling the exponential curve.

Navigating Through Points: Determining a and b

To fully define an exponential function, we need to determine the values of a and b. If we are given two points that lie on the graph of the function, we can use these points to create a system of equations and solve for a and b. This method leverages the fundamental relationship between the input and output of an exponential function.

The general strategy involves substituting the coordinates of the given points into the equation f(x) = a * b^x. This will yield two equations with two unknowns (a and b). We can then use algebraic techniques such as substitution or elimination to solve for the unknowns. The value of a can often be found directly from the y-intercept (the point where x = 0). Once a is known, we can substitute it into the other equation and solve for b. This process transforms the abstract representation of an exponential function into a concrete, usable model.

Decoding the Problem: A Step-by-Step Solution

In this specific problem, we are given that the exponential function f(x) = a * b^x passes through the points (0, 5) and (2, 20). Our mission is to determine the values of a and b. Let's embark on this mathematical journey step by step.

Step 1: Utilize the Point (0, 5)

Since the function passes through (0, 5), we know that f(0) = 5. Substituting x = 0 and f(0) = 5 into the exponential function f(x) = a * b^x, we get:

5 = a * b*⁰

Since any non-zero number raised to the power of 0 is 1, we have:

5 = a * 1

Therefore, a = 5. This immediately gives us the initial value of the exponential function.

Step 2: Harness the Power of (2, 20)

Next, we use the information from the point (2, 20), which tells us that f(2) = 20. Substituting x = 2 and f(2) = 20 into the exponential function f(x) = a * b^x, we get:

20 = a * b*²

We already know that a = 5, so we can substitute this value into the equation:

20 = 5 * b*²

Step 3: Unveiling the Value of b

To solve for b, we first divide both sides of the equation by 5:

4 = b²

Now, we take the square root of both sides:

b = ±2

However, in the context of exponential functions, the base b is usually considered to be positive (and not equal to 1). Therefore, we take the positive root:

b = 2

This reveals the growth factor of our exponential function.

Step 4: The Grand Finale

We have successfully determined the values of a and b:

• a = 5
• b = 2

Therefore, the exponential function that passes through the points (0, 5) and (2, 20) is f(x) = 5 * 2^x.

Real-World Applications: Exponential Functions in Action

Exponential functions are not confined to the realm of mathematics textbooks; they have profound implications in various real-world scenarios. Understanding these applications helps us appreciate the practical significance of exponential functions.

Population Growth

One of the most classic examples of exponential growth is in population dynamics. Under ideal conditions, a population can grow exponentially, doubling in size over a fixed period. This can be modeled using an exponential function where the base b represents the growth rate. For instance, the growth of bacteria in a petri dish or the population of a city can often be approximated by an exponential function, at least for a certain period.

Compound Interest

In the world of finance, compound interest is a prime example of exponential growth. When interest is compounded, it means that the interest earned in one period is added to the principal, and the next interest calculation is based on the new, higher principal. This compounding effect leads to exponential growth of the investment over time. The formula for compound interest is closely related to the general form of an exponential function.

Radioactive Decay

On the flip side, exponential decay is observed in radioactive materials. Radioactive isotopes decay over time, transforming into other elements. The rate of decay is proportional to the amount of the isotope present, leading to an exponential decrease in the amount of the isotope. This principle is used in radioactive dating, where the age of ancient artifacts and geological formations can be estimated by measuring the remaining amount of certain radioactive isotopes.

Learning Curves

In psychology and organizational behavior, exponential functions can model learning curves. These curves represent the rate at which a person learns a new skill or task. Initially, progress is rapid, but as the person becomes more proficient, the rate of learning slows down. This pattern can often be modeled using an exponential function with a base between 0 and 1, representing exponential decay in the rate of learning.

Conclusion: The Ubiquitous Exponential Function

In this exploration, we've unraveled the mysteries of exponential functions, demonstrating how to determine their parameters given points on their graph. We've seen how the values of a and b dictate the function's behavior, and we've explored a real-world example to solidify our understanding. Exponential functions are not just abstract mathematical concepts; they are powerful tools for modeling and understanding the world around us. From the growth of populations to the decay of radioactive substances, exponential functions provide a framework for making sense of a wide range of phenomena. By mastering the fundamentals of exponential functions, we unlock a deeper understanding of the dynamic systems that shape our world. Remember, the key to understanding exponential functions lies in grasping the significance of the base, b, and its impact on the function's growth or decay. The initial value, a, sets the stage, but it's b that orchestrates the exponential drama. As you continue your mathematical journey, keep an eye out for exponential functions in action, and you'll be amazed by their prevalence and power.

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