Exponential Function: Find F(x) From Points (0,2) & (2,8)

Hey guys! Let's dive into the fascinating world of exponential functions and figure out how to pinpoint the exact function when we're given a couple of points it passes through. Today, we're tackling a classic problem: finding the exponential function f(x) = ab^x given that its graph goes through the points (0, 2) and (2, 8). It might sound intimidating, but trust me, it's totally manageable, and we'll break it down step by step. Understanding exponential functions is crucial in various fields, from finance and biology to computer science. These functions model phenomena where growth or decay occurs at a rate proportional to the current amount. Think about population growth, compound interest, or radioactive decay – all these can be described using exponential functions. So, mastering the art of finding these functions from given data points is a valuable skill.

Understanding Exponential Functions

Before we jump into solving the problem, let's quickly recap what exponential functions are all about. An exponential function has the general form f(x) = ab^x, where:

• a is the initial value or the y-intercept (the value of f(x) when x = 0).
• b is the base, which determines the rate of growth (b > 1) or decay (0 < b < 1).
• x is the independent variable.

The key characteristic of exponential functions is that the dependent variable (f(x)) changes by a constant factor for each unit change in the independent variable (x). This constant factor is the base b. When b is greater than 1, the function grows exponentially; when b is between 0 and 1, the function decays exponentially. To truly grasp the essence of exponential functions, it's helpful to visualize their graphs. Exponential functions with a base greater than 1 exhibit rapid growth, soaring upwards as x increases. Conversely, those with a base between 0 and 1 decay, gradually approaching zero as x grows. This characteristic behavior makes them invaluable tools for modeling various real-world phenomena, from population dynamics to financial investments. Understanding the parameters a and b is crucial; a dictates the function's initial value, while b governs its growth or decay rate. This foundational knowledge sets the stage for tackling the problem at hand – determining the specific exponential function that passes through given points.

Setting up the Equations

Now that we have a solid understanding of exponential functions, let's use the given points (0, 2) and (2, 8) to find the values of a and b in our function f(x) = ab^x. We can do this by plugging the coordinates of each point into the equation, creating a system of two equations with two unknowns. This is a standard technique in algebra, allowing us to solve for the unknown parameters that define our function. The beauty of this approach lies in its systematic nature; by translating the graphical information (the points) into algebraic equations, we can leverage powerful tools to unravel the function's identity. Let's start with the point (0, 2). Substituting x = 0 and f(x) = 2 into the equation, we get:

2 = ab⁰

Since any number raised to the power of 0 is 1, this simplifies to:

2 = a * 1

Therefore:

a = 2

See? We've already found the value of a! This is a fantastic start, and it highlights the power of choosing the right starting point. The point (0, 2) was particularly helpful because it directly revealed the value of a, the initial value of our exponential function. Now, let's move on to the second point (2, 8). Substituting x = 2 and f(x) = 8 into the equation, we get:

8 = ab²

We now have two equations:

1. a = 2
2. 8 = ab²

We're one step closer to cracking this problem! We've successfully translated the given information into a manageable system of equations. Now, we can use the value of a we just found to solve for b. This process of substituting known values to find unknowns is a cornerstone of algebraic problem-solving, and it's exactly what we'll do in the next step.

Solving for 'b'

We've already determined that a = 2. Now, we can substitute this value into the second equation, 8 = ab², to solve for b. This is where the magic of substitution comes into play. By replacing a with its known value, we transform the equation into one with a single unknown, making it much easier to solve. This technique is a fundamental tool in algebra, and it's used extensively in solving various types of equations. So, let's do it:

8 = 2b²

To isolate b², we divide both sides of the equation by 2:

4 = b²

Now, to find b, we take the square root of both sides. Remember that taking the square root can result in both positive and negative solutions, but in the context of exponential functions, the base b is usually positive (and not equal to 1). So, we'll consider the positive root:

b = √4

Therefore:

b = 2

Awesome! We've found the value of b. This step was crucial, as b determines the growth or decay rate of our exponential function. In this case, b = 2 indicates exponential growth. We're now just one step away from fully defining our function. We have the values of both a and b, and all that's left is to plug them back into the general form of the exponential function. This final step will give us the specific function that passes through the given points.

The Solution

We've found that a = 2 and b = 2. Now, we simply substitute these values back into the general form of the exponential function, f(x) = ab^x, to get the specific function that passes through the points (0, 2) and (2, 8). This final step is the culmination of our efforts, bringing together the individual pieces we've discovered to form the complete solution. It's like putting the last pieces of a puzzle together, revealing the full picture. So, let's plug in the values:

f(x) = 2 * 2^x

And there you have it! This is the exponential function whose graph goes through the points (0, 2) and (2, 8). We've successfully navigated the problem, using our understanding of exponential functions and algebraic techniques to arrive at the solution. This process highlights the power of mathematical reasoning, allowing us to translate graphical information into precise algebraic expressions. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. We started by understanding the general form of an exponential function, then used the given points to create a system of equations, solved for the unknowns, and finally, constructed the specific function. This methodical approach can be applied to a wide range of mathematical problems, making it a valuable skill to cultivate. So, let's recap our findings:

• b = 2
• a = 2

And the exponential function is:

f(x) = 2 * 2^x

Great job, guys! We nailed it!