Overview

This topic explores how quantities change over time using linear and exponential models. You'll learn how to identify constant growth vs. percentage-based growth, write equations that model real-world scenarios, and interpret growth trends through tables, graphs, and equations.

Key Concepts and Structures

• Linear Growth: Adds a constant amount each time period. General form: y = mx + b
  • m = constant rate of change (slope)
  • b = starting value (initial amount)
• Exponential Growth: Multiplies by a constant factor each time period. General form: y = a(1 + r)^t
  • a = initial amount
  • r = growth rate (as a decimal)
  • t = time
• Exponential Decay: Uses the form y = a(1 - r)^t. Common in depreciation and population decline models.
• Growth vs. Decay: Growth means increasing over time; decay means decreasing. The base of the exponent determines the trend.
• Graph Comparison:
  • Linear: straight line, constant slope
  • Exponential: curves upward or downward, increasing or decreasing rapidly
• Real-World Examples: Linear — salary increases, savings with regular deposits. Exponential — population growth, interest, virus spread.

Step-by-Step Example

Problem: A population of 1,000 grows by 5% per year. What is the population after 3 years?

Step 1: Identify values: a = 1000, r = 0.05, t = 3

Step 2: Use exponential growth formula: y = a(1 + r)^t

Step 3: Plug in values:
y = 1000(1 + 0.05)^3 = 1000(1.157625) ≈ 1157.63

Final Answer: Approximately 1,158 people after 3 years.

Quick Tip