How to avoid errors writing exponential functions from two points

📚 Understanding Exponential Functions

An exponential function is a function of the form $f(x) = ab^x$, where $a$ is the initial value and $b$ is the growth or decay factor. When given two points, $(x_1, y_1)$ and $(x_2, y_2)$, our goal is to determine the values of $a$ and $b$ to define the specific exponential function that passes through these points.

📜 Historical Context

The study of exponential functions dates back to the development of calculus and the understanding of continuous growth and decay. Early applications included modeling population growth and radioactive decay. Leonhard Euler made significant contributions, solidifying the notation and properties of exponential functions.

⚗️ Key Principles

• 🧩 Understanding the General Form: Remember that an exponential function is given by $f(x) = ab^x$. Our goal is to find the values of $a$ and $b$.
• 🔢 Setting up Equations: Use the given points $(x_1, y_1)$ and $(x_2, y_2)$ to create two equations:
  • $y_1 = ab^{x_1}$
  • $y_2 = ab^{x_2}$
• ➗ Solving for $b$ by Division: Divide the second equation by the first equation to eliminate $a$: $\frac{y_2}{y_1} = \frac{ab^{x_2}}{ab^{x_1}} = b^{x_2 - x_1}$. Then, solve for $b$: $b = \sqrt[x_2-x_1]{\frac{y_2}{y_1}}$
• 🔄 Solving for $a$: Substitute the value of $b$ back into either of the original equations (e.g., $y_1 = ab^{x_1}$) and solve for $a$: $a = \frac{y_1}{b^{x_1}}$
• ✅ Verifying the Solution: Plug the values of $a$ and $b$ back into the general equation $f(x) = ab^x$, and check if both given points satisfy the equation.

💡 Common Errors and How to Avoid Them

• 🧮 Algebra Mistakes: Double-check your algebraic manipulations, especially when solving for $b$ and $a$. Use a calculator for complex calculations.
• ➕ Sign Errors: Pay close attention to signs, especially when dealing with negative exponents or negative values for $y_1$ and $y_2$.
• 📝 Incorrect Substitution: Ensure you substitute the correct values of $x$ and $y$ into the equations.
• ♾️ Undefined Cases: Be aware of cases where $y_1$ or $y_2$ are zero, as this can lead to undefined values or require special handling.
• 🖥️ Calculator Errors: When using a calculator, ensure that you enter the values correctly and use the correct functions (e.g., exponentiation, roots).

🧪 Real-world Examples

Example 1: Population Growth

Suppose a population of bacteria is observed at two points in time. At time $x_1 = 0$ hours, the population is $y_1 = 1000$. At time $x_2 = 2$ hours, the population is $y_2 = 4000$. Find the exponential function that models this growth.

1. Set up equations:
• $1000 = ab^0$
• $4000 = ab^2$
2. Solve for $b$: Since $b^0 = 1$, $a = 1000$. Then, $4000 = 1000b^2$, so $b^2 = 4$ and $b = 2$.
3. The exponential function is $f(x) = 1000(2)^x$.

Example 2: Radioactive Decay

A radioactive substance decays such that at time $x_1 = 0$ years, there are $y_1 = 50$ grams. At time $x_2 = 3$ years, there are $y_2 = 20$ grams. Find the exponential decay function.

1. Set up equations:
• $50 = ab^0$
• $20 = ab^3$
2. Solve for $b$: Since $b^0 = 1$, $a = 50$. Then, $20 = 50b^3$, so $b^3 = 0.4$ and $b = \sqrt[3]{0.4} \approx 0.7368$.
3. The exponential function is approximately $f(x) = 50(0.7368)^x$.

📝 Practice Quiz

Find the exponential function $f(x) = ab^x$ passing through the following points:

1. (0, 5) and (1, 10)
2. (1, 6) and (3, 24)
3. (0, 2) and (2, 50)

🌍 Real-World Applications

• 🏦 Finance: Modeling compound interest and investment growth.
• 🦠 Biology: Describing population growth, bacterial cultures, and the spread of diseases.
• ☢️ Physics: Understanding radioactive decay and the half-life of isotopes.
• 🌡️ Environmental Science: Modeling the spread of pollutants and resource depletion.

🔑 Conclusion

Mastering the process of finding exponential functions from two points involves careful algebraic manipulation, a solid understanding of the exponential form, and attention to detail. By avoiding common errors and practicing with real-world examples, you can confidently solve a wide range of problems involving exponential growth and decay. Remember to always verify your solutions to ensure accuracy!