Find An Exponential Equation With Two Points

Finding an Exponential Equation with Two Points: A Comprehensive Guide

Finding the equation of an exponential function given two points might seem daunting, but with a structured approach and a solid understanding of exponential functions, it becomes a manageable task. This comprehensive guide will walk you through the process, explaining the underlying principles and providing detailed examples to solidify your understanding.

Understanding Exponential Functions

Before diving into the process of finding the equation, let's revisit the fundamental form of an exponential function:

y = ab^x

Where:

• y represents the dependent variable.
• x represents the independent variable.
• a represents the initial value (the y-intercept, the value of y when x = 0).
• b represents the base, which determines the rate of growth or decay. If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay.

Our goal is to determine the values of 'a' and 'b' using the two given points.

The Two-Point Method: A Step-by-Step Approach

Let's assume we are given two points, (x₁, y₁) and (x₂, y₂). To find the equation, we'll follow these steps:

Step 1: Setting up the Equations

Substitute each point into the general exponential equation:

• y₁ = ab^x₁
• y₂ = ab^x₂

This gives us a system of two equations with two unknowns (a and b).

Step 2: Solving for 'b'

To solve for 'b', we can divide the second equation by the first:

(y₂/y₁) = (ab^x₂) / (ab^x₁)

Simplifying, we get:

(y₂/y₁) = b^{(x₂ - x₁)}

Now, take the logarithm (base 10 or natural logarithm, ln) of both sides:

log(y₂/y₁) = (x₂ - x₁)log(b)

Solving for 'b':

log(b) = log(y₂/y₁) / (x₂ - x₁)

b = 10^{[log(y₂/y₁) / (x₂ - x₁)]} (using base 10 logarithm)

Or, using natural logarithm:

ln(b) = ln(y₂/y₁) / (x₂ - x₁)

b = e^{[ln(y₂/y₁) / (x₂ - x₁)]}

Step 3: Solving for 'a'

Once we have the value of 'b', we can substitute it back into either of the original equations (from Step 1) to solve for 'a'. Let's use the first equation:

y₁ = ab^x₁

Solving for 'a':

a = y₁ / b^x₁

Step 4: Writing the Exponential Equation

Finally, substitute the values of 'a' and 'b' back into the general exponential equation:

y = ab^x

This is the exponential equation that passes through the two given points.

Detailed Examples

Let's work through a couple of examples to illustrate the process:

Example 1: Exponential Growth

Find the exponential equation that passes through the points (1, 6) and (3, 24).

1. Setting up the equations:

   • 6 = ab¹
   • 24 = ab³

2. Solving for 'b':

   • (24/6) = b^(3-1)
   • 4 = b²
   • b = 2 (We take the positive square root since b must be positive in an exponential function)

3. Solving for 'a':

   • 6 = a(2)¹
   • a = 3

4. Writing the equation:

   • y = 3(2)^x

Therefore, the exponential equation that passes through (1, 6) and (3, 24) is y = 3(2)^x.

Example 2: Exponential Decay

Find the exponential equation that passes through the points (0, 10) and (2, 2.5).

1. Setting up the equations:

   • 10 = ab⁰
   • 2.5 = ab²

2. Solving for 'b':

   • (2.5/10) = b^(2-0)
   • 0.25 = b²
   • b = 0.5 (Again, we take the positive square root)

3. Solving for 'a':

   • 10 = a(0.5)⁰
   • a = 10

4. Writing the equation:

   • y = 10(0.5)^x

Therefore, the exponential equation that passes through (0, 10) and (2, 2.5) is y = 10(0.5)^x.

Handling Special Cases

While the above method works for most cases, there are a few special scenarios to consider:

• One point has x = 0: If one of the points has an x-coordinate of 0, then the y-coordinate of that point directly gives you the value of 'a' (since b⁰ = 1). This simplifies the calculations significantly.

• Points lead to b = 1: If your calculations lead to b = 1, then the function is not truly exponential; it's a constant function (y = a). This indicates that the two given points do not define an exponential relationship.

• Negative values of y: Remember that the base 'b' must always be positive. If the y-values are negative, it might be necessary to adjust the model to consider a reflection or a different type of function.

Verifying Your Solution

After finding the equation, it's always a good practice to verify your solution by substituting the original points back into the equation. If both points satisfy the equation, you have successfully determined the correct exponential function.

Conclusion

Finding an exponential equation given two points is a valuable skill in various fields, from data analysis and modeling to finance and physics. By systematically applying the steps outlined in this guide, you can confidently solve these problems and gain a deeper understanding of exponential functions and their applications. Remember to always check your work and consider the special cases to ensure accuracy. Practice with different examples to build your proficiency and confidence.