Finding Exponential Function Formula Through Points (-1, 4/5) And (3, 500)

Finding the formula for an exponential function that passes through two given points is a fundamental problem in mathematics. Exponential functions are ubiquitous in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. This article will walk you through the process of determining the specific formula for an exponential function when provided with two points it passes through. We will explore the underlying principles, the step-by-step methodology, and provide a detailed example to solidify your understanding.

Understanding Exponential Functions

Exponential functions are characterized by their rapid growth or decay, making them crucial in modeling various natural and financial phenomena. The general form of an exponential function is given by:

$$f(x) = ab^x$$

Where:

• $f(x)$ is the value of the function at $x$.
• $a$ is the initial value or the y-intercept (the value of the function 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.

To define a specific exponential function, we need to determine the values of the constants $a$ and $b$. When given two points, we can set up a system of equations to solve for these unknowns. This process involves substituting the coordinates of the points into the general form and then using algebraic techniques to find $a$ and $b$.

Setting up the Equations

Suppose we are given two points, $(x_1, y_1)$ and $(x_2, y_2)$, that the exponential function passes through. We can substitute these points into the general form of the exponential function to create two equations:

1. $y_1 = ab^{x_1}$
2. $y_2 = ab^{x_2}$

These two equations form a system that we can solve for $a$ and $b$. The strategy is typically to eliminate one variable (either $a$ or $b$) by dividing one equation by the other. This eliminates $a$ and allows us to solve for $b$. Once we find $b$, we can substitute it back into one of the original equations to solve for $a$.

Solving for the Base ($b$)

To eliminate $a$, we can divide the second equation by the first equation:

$$\frac{y_2}{y_1} = \frac{ab^{x_2}}{ab^{x_1}}$$

The $a$ terms cancel out, leaving us with:

$$\frac{y_2}{y_1} = b^{x_2 - x_1}$$

Now, we can solve for $b$ by taking the $(x_2 - x_1)$-th root of both sides:

$$b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}}$$

This formula allows us to directly calculate the base $b$ using the coordinates of the two given points. Understanding this step is crucial because the base determines whether the function represents exponential growth (if $b > 1$) or decay (if $0 < b < 1$). The rate of this growth or decay is directly linked to the magnitude of $b$, influencing how steeply the function rises or falls.

Solving for the Initial Value ($a$)

Once we have found the base $b$, we can substitute it back into either of the original equations to solve for the initial value $a$. Let's use the first equation:

$$y_1 = ab^{x_1}$$

Solving for $a$, we get:

$$a = \frac{y_1}{b^{x_1}}$$

Alternatively, we could use the second equation:

$$y_2 = ab^{x_2}$$

Solving for $a$, we get:

$$a = \frac{y_2}{b^{x_2}}$$

Both methods should yield the same value for $a$. The initial value $a$ represents the y-intercept of the exponential function, providing a starting point for the exponential growth or decay. Knowing $a$ and $b$ allows us to completely define the exponential function.

Step-by-Step Methodology

To find the formula for an exponential function passing through two points, follow these steps:

1. Identify the two points: Let the points be $(x_1, y_1)$ and $(x_2, y_2)$.
2. Solve for the base ($b$): Use the formula:

   $$b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}}$$

3. Solve for the initial value ($a$): Substitute the value of $b$ and one of the points into the general equation $y = ab^x$ and solve for $a$. For example, using the point $(x_1, y_1)$:

   $$a = \frac{y_1}{b^{x_1}}$$

4. Write the exponential function: Substitute the values of $a$ and $b$ into the general form $f(x) = ab^x$.

This structured approach ensures that we systematically determine the parameters of the exponential function, minimizing errors and clarifying the process. Each step is crucial, and understanding the rationale behind each one aids in applying this method to various problems.

Detailed Example

Let's find the formula for the exponential function passing through the points $\left(-1, \frac{4}{5}\right)$ and $(3, 500)$.

1. Identify the points:

   • $(x_1, y_1) = \left(-1, \frac{4}{5}\right)$
   • $(x_2, y_2) = (3, 500)$

2. Solve for the base ($b$):

   $$b = \left(\frac{500}{\frac{4}{5}}\right)^{\frac{1}{3 - (-1)}} = \left(\frac{500 \cdot 5}{4}\right)^{\frac{1}{4}} = \left(\frac{2500}{4}\right)^{\frac{1}{4}} = (625)^{\frac{1}{4}} = 5$$

   The base $b$ is 5, indicating exponential growth since $b > 1$. This value dictates the rate at which the function increases, and in this case, it signifies a relatively rapid growth.

3. Solve for the initial value ($a$): Using the point $\left(-1, \frac{4}{5}\right)$:

   $$a = \frac{\frac{4}{5}}{5^{-1}} = \frac{4}{5} \cdot 5 = 4$$

   The initial value $a$ is 4, which means the y-intercept of the function is at the point (0, 4). This is the starting value from which the exponential function grows.

4. Write the exponential function:

   $$f(x) = 4 \cdot 5^x$$

   The final exponential function is $f(x) = 4 \cdot 5^x$. This function passes through both given points and accurately models the exponential relationship defined by these points. Verifying this result by plugging in the original points can confirm the correctness of the function.

Common Pitfalls and How to Avoid Them

When finding the formula for an exponential function through two points, there are several common pitfalls that can lead to errors. Being aware of these pitfalls and understanding how to avoid them can significantly improve accuracy and efficiency.

1. Incorrectly Identifying Points:

   • Pitfall: Swapping the $x$ and $y$ coordinates or mislabeling $(x_1, y_1)$ and $(x_2, y_2)$.
   • Solution: Double-check the coordinates and labels before substituting them into the formulas. Write down the points clearly and ensure that you are using the correct values for each variable.

2. Errors in Calculation:

   • Pitfall: Making mistakes when calculating the base $b$ or the initial value $a$, especially with fractional exponents and divisions.
   • Solution: Use a calculator to perform the calculations, and double-check each step. Pay close attention to the order of operations and the rules of exponents. Consider simplifying fractions and roots to reduce errors.

3. Dividing by Zero:

   • Pitfall: If $x_1 = x_2$, the formula for $b$ will involve division by zero, which is undefined.
   • Solution: If $x_1 = x_2$, the function is not exponential. It may be a linear function or undefined. Check the problem statement and the nature of the function required.

4. Incorrectly Substituting Values:

   • Pitfall: Substituting values into the wrong places in the formulas, particularly when solving for $a$ after finding $b$.
   • Solution: Ensure that you are using the correct values for $y_1$, $b$, and $x_1$ (or $y_2$, $b$, and $x_2$) when solving for $a$. Rewrite the formulas with the specific values before performing the calculations to minimize errors.

5. Misunderstanding Exponential Properties:

   • Pitfall: Applying exponent rules incorrectly, such as simplifying fractional exponents or negative exponents.
   • Solution: Review the properties of exponents. For example, $b^{\frac{1}{n}}$ is the $n$-th root of $b$, and $b^{-x} = \frac{1}{b^x}$. Ensure you understand and apply these rules correctly.

6. Not Verifying the Solution:

   • Pitfall: Failing to check if the final exponential function actually passes through the given points.
   • Solution: Substitute the original points into the final equation to verify that the equation holds true for both points. This is a crucial step to catch any errors made during the process.

By being mindful of these common pitfalls and employing the suggested solutions, you can significantly improve your ability to accurately find the formula for an exponential function passing through two points.

Conclusion

Finding the formula for an exponential function that passes through two points involves a systematic approach of setting up and solving equations. By understanding the general form of exponential functions, the significance of the base and initial value, and the step-by-step methodology, you can confidently tackle these problems. Remember to be meticulous with your calculations, and always verify your solution to ensure accuracy. This skill is essential in various fields, including mathematics, physics, finance, and biology, where exponential functions play a crucial role in modeling and analyzing real-world phenomena. Mastering this technique provides a solid foundation for further exploration of exponential relationships and their applications.