Find an Exponential Function with Two Points: The Easy Way

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24-10-2024

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Finding an exponential function that passes through two specific points can seem daunting at first, but it's actually quite straightforward once you break it down. In this guide, we'll walk you through the easy way to determine an exponential function based on two points, complete with clear steps and examples. Let's dive in! 🚀

Understanding the Exponential Function

An exponential function has the general form:

f(x) = a * b^x

Where:

• a is a constant that represents the initial value.
• b is the base of the exponential function, and it determines the rate of growth (if b > 1) or decay (if 0 < b < 1).

To find the parameters a and b, we need two points. Let's denote these points as:

• Point 1: (x1, y1)
• Point 2: (x2, y2)

Setting Up the Equations

Given the two points, we can set up two equations based on the exponential function:

1. y1 = a * b^x1
2. y2 = a * b^x2

Now we have a system of equations that we can solve for a and b.

Step-by-Step Solution

Step 1: Solve for a

From the first equation, we can express a in terms of b:

[ a = \frac{y1}{b^{x1}} ]

Step 2: Substitute a into the Second Equation

Now, substitute this expression for a into the second equation:

[ y2 = \frac{y1}{b^{x1}} * b^{x2} ]

This simplifies to:

[ y2 = y1 * b^{(x2 - x1)} ]

Step 3: Solve for b

Now we can isolate b:

[ b^{(x2 - x1)} = \frac{y2}{y1} ]

Taking the logarithm of both sides will allow us to solve for b:

[ x2 - x1 = \log_b\left(\frac{y2}{y1}\right) ]

Example

Let's apply this to a specific example. Suppose we have the following points:

• Point 1: (1, 2)
• Point 2: (3, 8)

1. Set up the equations

Using our points in the exponential function:

1. 2 = a * b^1
2. 8 = a * b^3

2. Express a in terms of b

From the first equation:

[ a = 2 / b ]

3. Substitute into the second equation

Substituting into the second equation:

[ 8 = \left( \frac{2}{b} \right) * b^3 ]

This simplifies to:

[ 8 = 2b^2 ]

4. Solve for b

Dividing both sides by 2:

[ 4 = b^2 ]

Taking the square root gives:

[ b = 2 ]

5. Find a

Now substitute b back to find a:

[ a = \frac{2}{b} = \frac{2}{2} = 1 ]

Conclusion

The exponential function that passes through the points (1, 2) and (3, 8) is:

[ f(x) = 1 \cdot 2^x ]

Summary Table

Here's a quick reference table summarizing the process:

Now that you know how to find an exponential function with two points, you can apply this technique to any pair of points! Happy calculating! 🎉