How To Find Exponential Function From Two Points

Let's explore the fascinating world of exponential functions and delve into a practical skill: finding the specific exponential function that passes through two given points. This skill is invaluable in various fields, from modeling population growth and radioactive decay to understanding financial investments and the spread of information.

Understanding Exponential Functions

At its core, an exponential function describes a relationship where a quantity increases or decreases at a constant percentage rate over time. The general form of an exponential function is:

f(x) = a * b^x

Where:

• f(x) represents the value of the function at a given input x.
• a is the initial value or the y-intercept (the value of the function when x = 0).
• b is the base or growth/decay factor, determining whether the function increases (b > 1) or decreases (0 < b < 1).
• x is the independent variable, often representing time.

The key characteristic of an exponential function is the variable x appearing as an exponent. This leads to rapid growth when b > 1 and rapid decay when 0 < b < 1.

Why Find an Exponential Function Through Two Points?

In many real-world scenarios, we don't have the explicit equation of an exponential function. Instead, we might have data points representing the value of a quantity at different times. For example:

• The population of a city in 2010 and 2020.
• The remaining amount of a radioactive substance after 1 year and 5 years.
• The value of an investment after 2 years and 7 years.

Given these two points, our goal is to determine the values of a and b in the exponential function f(x) = a * b^x so that the function accurately models the relationship between x and f(x).

The Step-by-Step Process

Let's outline the procedure for finding an exponential function that passes through two given points (x₁, y₁) and (x₂, y₂).

Step 1: Set up a System of Equations

Substitute the coordinates of each point into the general form of the exponential function:

• Equation 1: y₁ = a * b^(x₁)
• Equation 2: y₂ = a * b^(x₂)

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

Step 2: Solve for 'a' in One of the Equations

Choose either Equation 1 or Equation 2 and solve for a. It's often easier to choose the equation with the simpler exponents. Let's solve Equation 1 for a:

• a = y₁ / b^(x₁)

Step 3: Substitute the Expression for 'a' into the Other Equation

Substitute the expression for a obtained in Step 2 into the other equation (the one you didn't use in Step 2). This will eliminate a and leave you with an equation in terms of b only.

• y₂ = (y₁ / b^(x₁)) * b^(x₂)

Step 4: Solve for 'b'

Simplify the equation and solve for b. This usually involves using exponent rules and algebraic manipulation.

• y₂ = y₁ * b^(x₂ - x₁)
• y₂ / y₁ = b^(x₂ - x₁)

Now, take the (x₂ - x₁)th root of both sides:

• b = (y₂ / y₁)^(1 / (x₂ - x₁))

Step 5: Substitute the Value of 'b' Back into the Expression for 'a'

Substitute the value of b you found in Step 4 back into the expression for a that you derived in Step 2.

• a = y₁ / b^(x₁)

Step 6: Write the Exponential Function

Now that you have found the values of a and b, write the exponential function in the form:

• f(x) = a * b^x

Example

Let's find the exponential function that passes through the points (2, 12) and (5, 324).

Step 1: Set up a System of Equations

• Equation 1: 12 = a * b²
• Equation 2: 324 = a * b⁵

Step 2: Solve for 'a' in Equation 1

• a = 12 / b²

Step 3: Substitute into Equation 2

• 324 = (12 / b²) * b⁵

Step 4: Solve for 'b'

• 324 = 12 * b³
• 324 / 12 = b³
• 27 = b³
• b = ³√27 = 3

Step 5: Substitute 'b' back into the expression for 'a'

• a = 12 / 3²
• a = 12 / 9
• a = 4/3

Step 6: Write the Exponential Function

• f(x) = (4/3) * 3^x

Therefore, the exponential function that passes through the points (2, 12) and (5, 324) is f(x) = (4/3) * 3^x.

Handling Special Cases and Considerations

While the above steps provide a general method, some special cases and considerations might arise.

• One of the y-values is zero: If either y₁ or y₂ is zero, and the other is not, then no exponential function of the form f(x) = a * b^x can pass through both points, assuming x₁ and x₂ are finite. Exponential functions of this form will never equal zero for finite values of x. You would need to consider a different type of function. If both y₁ and y₂ are zero, then any exponential function will technically work, but this is likely not what you intended.
• y₁ and y₂ have opposite signs: An exponential function of the form f(x) = a * b^x, where a and b are real numbers, cannot have both positive and negative y-values. This is because b^x will always be positive for any real number x if b is positive. And if a is positive, a * b^x* will always be positive. You'd need to consider another type of function, or allow for complex values of a or b.
• x₁ = x₂: If the x-values are the same, you don't have two distinct points, and you can't uniquely define an exponential function using this method. You would need additional information.
• Dealing with messy numbers: In some cases, the values of a and b might be irrational or difficult to simplify. Use a calculator to approximate the values as needed. The important thing is to follow the steps correctly.
• Checking your answer: After finding the exponential function, always plug the original points (x₁, y₁) and (x₂, y₂) back into the function to verify that the equation holds true. This helps catch any errors in your calculations.

Logarithmic Approach: An Alternative Perspective

While the direct substitution method is effective, understanding the logarithmic approach can provide valuable insights and sometimes simplify calculations.

The exponential function f(x) = a * b^x can be rewritten in logarithmic form. Taking the natural logarithm (ln) of both sides, we get:

• ln(f(x)) = ln(a * b^x)
• ln(f(x)) = ln(a) + x * ln(b)

Notice that this equation is now in the form of a linear equation:

• y = mx + c

Where:

• y = ln(f(x))
• m = ln(b) (the slope)
• x = x
• c = ln(a) (the y-intercept)

Therefore, finding an exponential function through two points is equivalent to finding a linear function through the points (x₁, ln(y₁)) and (x₂, ln(y₂)).

Using the Logarithmic Approach:

1. Transform the points: Take the natural logarithm of the y-values of your two points: (x₁, ln(y₁)) and (x₂, ln(y₂)).

2. Find the slope (m): Calculate the slope of the line passing through these two transformed points:

   • m = (ln(y₂) - ln(y₁)) / (x₂ - x₁)

3. Solve for 'b': Since m = ln(b), we can find b by taking the exponential of m:

   • b = e^m = e^((ln(y₂) - ln(y₁)) / (x₂ - x₁))
   • This simplifies to b = (y₂ / y₁)^(1 / (x₂ - x₁)), which is what we found before.

4. Find the y-intercept (c): Use the point-slope form of a linear equation to find the y-intercept c (which is ln(a)):

   • ln(y) - ln(y₁) = m(x - x₁)
   • ln(a) = ln(y₁) - m * x₁

5. Solve for 'a': Since c = ln(a), we can find a by taking the exponential of c:

   • a = e^c = e^(ln(y₁) - m * x₁) = y₁ / e^(m * x₁) = y₁ / b^(x₁)
   • Which is the same result as before.

6. Write the Exponential Function: Now that you have found the values of a and b, write the exponential function in the form:

   • f(x) = a * b^x

The logarithmic approach provides a different perspective on the problem and reinforces the connection between exponential and logarithmic functions. It also highlights how exponential relationships can be transformed into linear relationships, making them easier to analyze in certain contexts.

Applications and Real-World Examples

Finding exponential functions from two points has numerous applications in various fields.

• Population Growth: If you know the population of a city at two different points in time, you can find an exponential function to model its growth and predict future populations.
• Radioactive Decay: Knowing the amount of a radioactive substance at two different times allows you to determine its decay rate and predict its future levels. This is crucial in fields like nuclear medicine and environmental science.
• Financial Investments: You can model the growth of an investment using an exponential function if you know its value at two different times. This helps in understanding compound interest and predicting future returns.
• Spread of Information: The spread of a rumor or a virus can often be modeled using an exponential function. Knowing the number of people infected or aware of the rumor at two different times allows you to estimate the rate of spread and predict its future reach.
• Drug Dosage: The concentration of a drug in the bloodstream decays exponentially over time. Knowing the concentration at two different times helps determine the drug's elimination rate and optimize dosage schedules.
• Machine Learning and Data Analysis: Exponential functions are used in various machine learning algorithms, such as exponential smoothing for time series forecasting and in defining activation functions in neural networks.

Tips for Success

• Accuracy: Pay close attention to detail and perform calculations carefully to avoid errors.
• Understanding Exponent Rules: A solid understanding of exponent rules is crucial for simplifying equations and solving for b.
• Calculator Proficiency: Learn how to use your calculator effectively to evaluate exponents and logarithms.
• Practice: The more you practice, the more comfortable you will become with the process.
• Double-Check: Always verify your solution by plugging the original points back into the function.
• Consider the Context: Think about the real-world context of the problem and whether an exponential model is appropriate. In some cases, other types of functions might be more suitable.

Conclusion

Finding an exponential function that passes through two given points is a valuable skill with wide-ranging applications. By following the step-by-step procedure outlined in this article, understanding the underlying principles, and practicing regularly, you can master this technique and apply it to solve real-world problems. Whether you're modeling population growth, analyzing financial investments, or exploring the spread of information, the ability to find exponential functions from two points will empower you to make informed decisions and gain deeper insights into the world around you. Embrace the power of exponential functions and unlock their potential to model and predict the future.