How To Find Equation Of Exponential Graph With Two Points

How to Find the Equation of an Exponential Graph with Two Points

Finding the equation of an exponential graph given two points might seem daunting, but with a systematic approach, it's a straightforward process. This article will guide you through the steps, explaining the underlying concepts and providing practical examples. We'll explore how to determine the equation in the form y = abˣ, where 'a' is the initial value and 'b' is the base. Understanding this will help you analyze exponential growth and decay scenarios effectively.

Understanding Exponential Functions

An exponential function is characterized by its constant growth or decay rate. The general form is y = abˣ, 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 (b > 1) or decay (0 < b < 1).

Steps to Find the Equation

Let's assume we have two points, (x₁, y₁) and (x₂, y₂), that lie on the exponential graph. To find the equation, follow these steps:

1. Substitute the points into the general equation: This will give you two equations with two unknowns (a and b).

2. Solve for 'a' and 'b': There are several methods to solve this system of equations. One common approach involves dividing one equation by the other to eliminate 'a', then solving for 'b'. After finding 'b', substitute it back into either equation to solve for 'a'.

3. Write the equation: Once you have determined the values of 'a' and 'b', substitute them back into the general form y = abˣ to get the specific equation for your exponential graph.

Example: Finding the Equation

Let's say we have the points (1, 6) and (3, 24). Let's find the equation of the exponential function.

1. Substitute the points:

   • 6 = ab¹ (from point (1, 6))
   • 24 = ab³ (from point (3, 24))

2. Solve for 'a' and 'b': Divide the second equation by the first:

   • (24 = ab³) / (6 = ab¹) = 4 = b²
   • Therefore, b = 2 (We choose the positive value because exponential functions generally deal with positive growth or decay rates).

   Now, substitute b = 2 into the first equation:

   • 6 = a(2)¹
   • a = 3

3. Write the equation: The equation of the exponential graph is y = 3(2ˣ).

Dealing with Decay:

If the y-values decrease as x increases, you'll have an exponential decay function. The process remains the same, but the value of 'b' will be between 0 and 1.

Using Logarithms (for more complex scenarios):

For more complex scenarios or when dealing with decimal values, using logarithms can simplify the calculations. You can take the logarithm of both sides of your equations to solve for 'a' and 'b' more easily.

Conclusion:

Finding the equation of an exponential graph given two points is a manageable task using a step-by-step approach. Remember the general form y = abˣ, substitute your points, solve for 'a' and 'b', and then write the final equation. Understanding this process allows for a deeper analysis of exponential growth and decay models in various applications, from population growth to radioactive decay. By mastering this technique, you’ll gain valuable skills in mathematical modeling and data analysis.