Find A And B: Exponential Function Through Two Points

When we talk about exponential functions, we're often looking at patterns of growth or decay. These functions have a general form, f(x) = a * b^x, where 'a' and 'b' are constants we need to figure out. 'a' usually represents the initial value (when x=0), and 'b' is the growth or decay factor. Today, we're going to tackle a common problem: finding the values of 'a' and 'b' when you know two specific points that the exponential function passes through. This is a fundamental skill in understanding how exponential models work and how to apply them to real-world scenarios. Let's dive into an example where our function f(x) = a * b^x passes through the points (0, 10) and (3, 1250). Our goal is to determine the exact values of 'a' and 'b' that make this function true for both these points. This process involves a bit of algebra, but by breaking it down step-by-step, you'll see how straightforward it can be to unlock the secrets of these powerful functions.

Understanding the General Form of an Exponential Function

The general form of an exponential function, f(x) = a * b^x, is the cornerstone of our problem. Here, 'a' is the y-intercept, meaning it's the value of the function when x = 0. Think of it as your starting point. If you're modeling population growth, 'a' would be the initial population. If you're looking at radioactive decay, 'a' might be the initial amount of the substance. The 'b' term is known as the base or the growth/decay factor. If 'b' is greater than 1, the function exhibits exponential growth – it increases rapidly as 'x' increases. If 'b' is between 0 and 1, the function shows exponential decay – it decreases rapidly as 'x' increases. If 'b' is equal to 1, the function becomes a constant f(x) = a, which isn't typically considered exponential. The exponent 'x' determines how many times the base 'b' is multiplied by itself. In our specific problem, we are given that the function f(x) = a * b^x passes through two points: (0, 10) and (3, 1250). Each point gives us a pair of (x, y) values that must satisfy the equation of our function. This means we can substitute these x and y values into f(x) = a * b^x to create a system of equations that we can then solve for 'a' and 'b'. This approach allows us to precisely define the unique exponential function that fits the given data points, making it a versatile tool for modeling various phenomena.

Using the First Point: (0, 10)

Let's start with the first point provided: (0, 10). This point tells us that when x = 0, the value of the function f(x) is 10. We can substitute these values directly into our general exponential function f(x) = a * b^x. So, we have:

10 = a * b^0

Now, remember a fundamental rule in mathematics: any non-zero number raised to the power of zero is equal to 1. So, b^0 = 1. Substituting this back into our equation, we get:

10 = a * 1

This simplifies to:

a = 10

This is a significant finding! We've just determined the value of 'a' using only the first point. In an exponential function of the form f(x) = a * b^x, the coefficient 'a' always represents the y-intercept, which is the value of the function when x = 0. Therefore, if a point with an x-coordinate of 0 is given, the y-coordinate of that point is the value of 'a'. This makes solving for 'a' incredibly straightforward in such cases. Now that we know a = 10, our exponential function takes on a more specific form: f(x) = 10 * b^x. We still need to find the value of 'b', and for that, we'll use the second point.

Using the Second Point: (3, 1250)

Now we move on to our second point: (3, 1250). This means when x = 3, f(x) = 1250. We already know that a = 10, so we can substitute both these values into our partially defined function f(x) = 10 * b^x:

1250 = 10 * b^3

Our next step is to isolate b^3. To do this, we divide both sides of the equation by 10:

1250 / 10 = b^3

125 = b^3

Now, we need to find the value of 'b'. This means we need to find the cube root of 125. In other words, we're looking for a number that, when multiplied by itself three times, equals 125.

We can think of it like this:

b * b * b = 125

If we try a few numbers, we might realize that 5 * 5 * 5 = 25 * 5 = 125.

Therefore, the cube root of 125 is 5.

b = 5

Congratulations! We have successfully found the value of 'b'. We now have both the value of 'a' and the value of 'b'.

The Complete Exponential Function

With the values we've found, a = 10 and b = 5, we can now write the complete, specific exponential function that passes through both the points (0, 10) and (3, 1250). The function is:

f(x) = 10 * 5^x

This equation precisely describes the relationship between x and f(x) for the given conditions. We can quickly verify this by plugging our original points back into the equation:

For the point (0, 10): f(0) = 10 * 5^0 = 10 * 1 = 10. This matches!

For the point (3, 1250): f(3) = 10 * 5^3 = 10 * (5 * 5 * 5) = 10 * 125 = 1250. This also matches!

The power of this method lies in its systematic approach. By utilizing the properties of exponents, especially b^0 = 1, and by treating the given points as equations, we can systematically solve for the unknown parameters 'a' and 'b'. This technique is not only applicable to this specific problem but is a foundational method for fitting exponential models to data in various fields, from finance and biology to physics and engineering. Understanding how to determine the constants of an exponential function is crucial for predicting future values, analyzing trends, and making informed decisions based on exponential growth or decay patterns. Whether you're a student learning algebra or a professional working with data, mastering this skill will undoubtedly enhance your analytical capabilities.

Why This Matters in Real-World Applications

Understanding how to find the specific parameters 'a' and 'b' for an exponential function like f(x) = a * b^x is far more than just an academic exercise; it's a vital skill for comprehending and modeling numerous real-world phenomena. For instance, in biology, it's used to model population growth of bacteria, animals, or even the spread of diseases. The 'a' value would represent the initial population or the initial number of infected individuals, while 'b' would describe the rate at which the population or infection spreads. In finance, exponential functions are fundamental to calculating compound interest. The initial investment is 'a', and 'b' relates to the annual interest rate. Understanding these functions allows investors to predict the future value of their investments. Physics employs exponential functions to describe processes like radioactive decay (where 'b' is less than 1, indicating a decrease in the amount of a substance over time) or the cooling of an object. Technology also relies on exponential growth, such as the increasing processing power of computers (often described by Moore's Law, which historically showed exponential growth) or the spread of information on the internet. In essence, any situation where a quantity changes at a rate proportional to its current value can be modeled using an exponential function. By being able to determine 'a' and 'b' from known data points, we gain the ability to create accurate predictive models, analyze past trends, and make informed decisions about the future. It empowers us to understand complex systems and to forecast outcomes with greater precision. The ability to solve for 'a' and 'b' unlocks the predictive power of exponential functions, making them indispensable tools in our data-driven world.

Conclusion

We've successfully navigated the process of finding the constants a and b in an exponential function f(x) = a * b^x by using two given points: (0, 10) and (3, 1250). We found that a = 10 and b = 5, resulting in the specific function f(x) = 10 * 5^x. This method, which leverages the property that b^0 = 1 and solves a system of equations, is a powerful technique applicable across many scientific and financial disciplines. Understanding these exponential relationships is key to interpreting growth and decay patterns in the world around us. For further exploration into the fascinating world of exponential functions and their applications, you might find resources from Khan Academy or Brilliant.org to be incredibly helpful.