Exponential Function Analysis

Problem Statement

• Determine an exponential function passing through points (-1, 5/4) and (2, 80).
• Find the exponential growth or decay rate.
• Calculate the function value when x = 3.

Theoretical Foundation

• Exponential Function Form:
  • General form: f(x) = a * b^x
  • Base (b): Equals 1 + r, where r is the exponential growth/decay rate.
  • a: Initial value or function value at x = 0.*

Given Points Analysis

• Point (-1, 5/4):
  • f(-1) = 5/4 → Equation: a * b^(-1) = 5/4
• Point (2, 80):
  • f(2) = 80 → Equation: a * b^2 = 80

Solving for a and b

1. From Point (-1, 5/4):
   • a * b^(-1) = 5/4
   • Solve for a: a = 5/4 * b
2. Substitute into Point (2, 80):
   • Substitute a = 5/4 * b into a * b^2 = 80
   • Equation: (5/4 * b) * b^2 = 80
   • Simplifies to: 5/4 * b^3 = 80
3. Solve for b:
   • Multiply both sides by 4/5: b^3 = 64
   • Take the cube root: b = 4*

Finding Initial Value (a)

• Use f(2) = 80:
  • a * 4^2 = 80
  • a * 16 = 80
  • Solve: a = 5

Exponential Function

• Final Function:
  • f(x) = 5 * 4^x*

Exponential Growth Rate

• Base b = 4:
  • b = 1 + r → 4 = 1 + r
  • Solve for r: r = 3
  • Growth Rate: 3 = 300%

Function Value at x = 3

• f(3):
  • f(3) = 5 * 4^3
  • Calculate: 4^3 = 64
  • Result: f(3) = 5 * 64 = 320

Conclusion

• Exponential Function: f(x) = 5 * 4^x
• Growth Rate: 300%
• Value at x = 3: 320*

The lecture demonstrates how to derive an exponential function, find its growth rate, and calculate specific function values using given points. It emphasizes solving a system of equations and verifying the function through the graph.