Guide to Exponential Growth and Decay Functions

Exponential Growth & Decay

Step-by-Step Guide to Understanding Real-World Functions

What is Exponential Growth?

Exponential growth occurs when quantity increases by the same percentage (or factor) in each unit of time. The curve starts slowly and then shoots upward rapidly.

Key Features:<ul><li>The base (b) is greater than 1.</li><li>Graph curves upwards from left to right.</li><li>Example: Population growth, compound interest.</li></ul>

What is Exponential Decay?

Exponential decay occurs when a quantity decreases by the same percentage over time. It drops rapidly at first and then levels off.

Key Features:<ul><li>The base (b) is between 0 and 1.</li><li>Graph curves downwards from left to right.</li><li>Example: Radioactive decay, car depreciation.</li></ul>

The General Formula

y = a(1 ± r)^t

<ul><li><strong>y</strong> = Final amount</li><li><strong>a</strong> = Initial amount (Principal)</li><li><strong>r</strong> = Growth or Decay rate (decimal)</li><li><strong>t</strong> = Time period</li><li>Use <strong>(1 + r)</strong> for Growth</li><li>Use <strong>(1 - r)</strong> for Decay</li></ul>

Step-by-Step Example 1

Scenario: A bacteria culture starts with 500 bacteria and grows at a rate of 12% per hour. How many bacteria will there be after 6 hours?

1. Identify variables: a=500, r=0.12, t=6<br>2. Determine type: Growth (use +)<br>3. Write equation: y = 500(1 + 0.12)^6<br>4. Solve: y = 500(1.12)^6 ≈ 987

Step-by-Step Example 2

Scenario: You bought a car for $25,000. It depreciates in value by 10% each year. write the function for the car's value.

1. Identify variables: a=25,000, r=0.10<br>2. Determine type: Decay (use -)<br>3. Base factor: (1 - 0.10) = 0.90<br>4. Function: y = 25,000(0.90)^t

Practice Problems: Table 1

An investment of $10,000 gains 8% interest per year.

Exponential Growth

y = 10,000(1.08)^t

Value increases over time.

A town of 5,000 people loses 2% of its population annually.

Exponential Decay

y = 5,000(0.98)^t

Value decreases over time.

Practice Problems: Table 2

The value of a rare coin ($500) appreciates by 4.5% yearly.

Exponential Growth

y = 500(1.045)^t

Base is 1.045

Amount of medicine (200mg) in the blood decreases by 20% per hour.

Exponential Decay

y = 200(0.80)^t

Base is 0.80

Visualizing the Difference

Comparison of Growth (10% rate) vs Decay (10% rate) starting at 100 units over 10 years.

Key Takeaways

Formula: y = a(1 ± r)^t

Growth: Base is > 1. Value increases.

Decay: Base is between 0 and 1. Value decreases.