What is an exponential function given two points?

📚 Understanding Exponential Functions Given Two Points

An exponential function has the form $f(x) = ab^x$, where $a$ is the initial value and $b$ is the base (or growth/decay factor). When given two points, our goal is to find the values of $a$ and $b$ that satisfy both points.

📜 Historical Context

Exponential functions have been used for centuries to model phenomena that exhibit rapid growth or decay. From compound interest calculations in ancient Babylon to population growth models developed by Thomas Malthus, the concept of exponential change has been a cornerstone of mathematical modeling.

🔑 Key Principles

• 📍 Point Representation: Each given point is represented as $(x, y)$, where $x$ is the independent variable and $y$ is the dependent variable.
• ✍️ Equation Setup: Substitute the $x$ and $y$ values of each point into the general form $y = ab^x$ to create two equations.
• ➗ Solving for 'b': Divide one equation by the other to eliminate $a$ and solve for $b$. This often involves taking roots.
• ➕ Solving for 'a': Substitute the value of $b$ back into one of the original equations to solve for $a$.
• ✅ Verification: Check your values of $a$ and $b$ by plugging them back into both original equations to ensure they hold true.

📈 Practical Steps to Find the Exponential Function

1. 📍 Identify the Points: Let's say we have two points, $(x_1, y_1)$ and $(x_2, y_2)$.
2. ✍️ Create Equations: Using the general form $y = ab^x$, create two equations:
   • Equation 1: $y_1 = ab^{x_1}$
   • Equation 2: $y_2 = ab^{x_2}$
3. ➗ Divide the Equations: Divide Equation 2 by Equation 1:
   • $\frac{y_2}{y_1} = \frac{ab^{x_2}}{ab^{x_1}}$
   • Simplify: $\frac{y_2}{y_1} = b^{x_2 - x_1}$
4. ⎚️ Solve for 'b': Take the $(x_2 - x_1)$-th root of both sides:
   • $b = \sqrt[x_2 - x_1]{\frac{y_2}{y_1}}$
5. ➕ Solve for 'a': Substitute the value of $b$ into either Equation 1 or Equation 2. Let's use Equation 1:
   • $y_1 = ab^{x_1}$
   • $a = \frac{y_1}{b^{x_1}}$
6. ✅ Write the Function: Now that you have $a$ and $b$, write the exponential function: $f(x) = ab^x$

⚙️ Real-World Examples

Example 1: Bacterial Growth

Suppose a bacterial culture has 200 bacteria initially. After 2 hours, there are 3200 bacteria. Find the exponential function representing the growth.

We have two points: $(0, 200)$ and $(2, 3200)$.

1. Write the equations:
   • $200 = ab^0$
   • $3200 = ab^2$
2. Solve for $a$:
   • From the first equation, $a = 200$ (since $b^0 = 1$).
3. Substitute $a$ into the second equation:
   • $3200 = 200b^2$
   • $b^2 = 16$
   • $b = 4$ (since growth implies $b > 0$).
4. The exponential function is $f(x) = 200(4)^x$.

Example 2: Radioactive Decay

A radioactive substance decays from 100 grams to 25 grams in 10 years. Find the exponential function representing the decay.

We have two points: $(0, 100)$ and $(10, 25)$.

1. Write the equations:
   • $100 = ab^0$
   • $25 = ab^{10}$
2. Solve for $a$:
   • From the first equation, $a = 100$ (since $b^0 = 1$).
3. Substitute $a$ into the second equation:
   • $25 = 100b^{10}$
   • $b^{10} = 0.25$
   • $b = \sqrt[10]{0.25} \approx 0.8909$
4. The exponential function is $f(x) = 100(0.8909)^x$.

✍️ Practice Quiz

1. ❓ Find the exponential function passing through $(1, 6)$ and $(3, 24)$.
2. ❓ Find the exponential function passing through $(0, 5)$ and $(2, 20)$.
3. ❓ Find the exponential function passing through $(-1, 2)$ and $(1, 8)$.

💡 Conclusion

Finding an exponential function given two points involves setting up and solving a system of equations. By understanding the underlying principles and following the steps outlined above, you can confidently tackle these problems. Remember to verify your results and apply these techniques to real-world scenarios to deepen your understanding.