Equations of Exponential Functions

Equations of Exponential Functions

Learning Outcomes

• Understand how to model exponential functions using data points.

• Derive exponential functions from given points and graphs.

• Utilize graphing calculators to evaluate exponential functions.

• Apply exponential models to continuous growth or decay scenarios.

Key Concepts

• Exponential Function: Generally in the form of f(x) = ab^x where:

  • a = initial value

  • b = growth/decay factor

Writing an Exponential Model

Step-by-Step Guide

1. Identify the Initial Value

   • If a data point is of the form (0, a), then a is the initial value.

2. Substituting Data Points

   • For two points (0, a) and (x, f(x)), use f(x) = ab^x to set up equations.

3. Solve for Growth/Decay Factor

   • Rearrange and solve for b.

Example: Deer Population

1. Initial data points: (0, 80) and (6, 180)

2. Set the model: N(t) = 80b^t

3. Substitute to find b:

   • For t=6: 180 = 80b^6

   • b^6 = \frac{180}{80} = \frac{9}{4}

   • b = \left(\frac{9}{4}\right)^{\frac{1}{6}} \approx 1.1447

4. Final model:

   • N(t) = 80(1.1447)^t

   • Population growth observed on the graph shows exponential behavior.

Finding Models Without Initial Value

• When the initial value is not known, use two data points to solve.

  • Example points: (-2, 6) and (2, 1) could be used.

    • Set equations:

      • 6 = ab^{-2}

      • 1 = ab^{2}

  • Solve simultaneously to find a and b.

Continuous Growth Models

• Form: A(t) = ae^{rt} where:

  • a = initial amount

  • r = growth rate (positive for growth, negative for decay)

  • t = time period

Example Calculations

1. Investment Problem:

   • Initial amount: P = 1000; annual interest: r = 0.10; time t = 1 year.

   • A(t) = 1000e^{0.10 \times 1} \approx 1105.17

2. Radiation Decay:

   • Initial amount of Radon: 100 mg, decay rate: -0.173 per day; find remaining after 3 days.

   • A(3) = 100e^{-0.173 \times 3} \approx 59.5115 mg

Conclusion & Important Notes

• Ensure correct identification of model types (exponential growth/decay).

• Always check the nature of the data points to validate if they represent exponential behavior.

• Use software tools where necessary for precise calculations and graphical representation.