# Guide to Exponential Growth and Decay Functions
> Learn to calculate exponential growth and decay using real-world examples like compound interest, population, and depreciation with step-by-step formulas.

Tags: exponential-growth, exponential-decay, mathematics, algebra-functions, growth-rate-formula, educational-presentation
## Exponential Growth & Decay: Real-World Functions

## What is Exponential Growth?
* Definition: Quantity increases by the same percentage in each unit of time.
* Key Features: Base (b) > 1, graph curves upward.
* Examples: Population growth, compound interest.

## What is Exponential Decay?
* Definition: Quantity decreases by the same percentage over time.
* Key Features: Base (b) between 0 and 1, graph curves downward.
* Examples: Radioactive decay, car depreciation.

## The General Formula
* **y = a(1 ± r)^t**
* **y**: Final amount
* **a**: Initial amount
* **r**: Rate (decimal)
* **t**: Time

## Step-by-Step Examples
* **Example 1 (Growth)**: Bacteria culture starting at 500 growing at 12% for 6 hours: y = 500(1.12)^6 ≈ 987.
* **Example 2 (Decay)**: Car bought for $25,000 depreciating at 10%: y = 25,000(0.90)^t.

## Practice Problems
* Investment: $10,000 at 8% growth → y = 10,000(1.08)^t
* Town Population: 5,000 losing 2% annually → y = 5,000(0.98)^t
* Rare Coin: $500 appreciating at 4.5% → y = 500(1.045)^t
* Medicine: 200mg decreasing by 20% hourly → y = 200(0.80)^t

## Visualizing the Difference
Comparison shows a 10% growth rate climbing to 259 units vs a 10% decay rate falling to 34 units over a 10-year period starting from 100 units.

## Key Takeaways
* Growth base > 1.
* Decay base < 1.
* Shared formula: y = a(1 ± r)^t.
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