Exponentials

SAT Math — Exponential Functions and Percent Change ## What Is an Exponential Function? In a linear function, the output changes by a constant amount for every unit increase in x (you add or subtract the same number). In an exponential function, the output changes by a constant factor (you multiply or divide by the same number). Standard form: y = a · bˣ - a = initial value (the value when x = 0) - b = growth/decay factor - b > 1: Exponential growth (value increases) - 0 < b < 1: Exponential decay (value decreases) ## Exponential Growth Example: A population of 500 bacteria doubles every hour. - After 0 hours: 500 - After 1 hour: 1,000 - After 2 hours: 2,000 - Equation: P = 500 · 2ʰ Reading the equation: - a = 500: Starting population - b = 2: Population doubles each hour (growth factor) ## Exponential Decay Example: A radioactive substance starts at 200 grams and loses 10% of its mass each year. - Each year, 90% remains: b = 1 − 0.10 = 0.90 - Equation: M = 200 · (0.90)ʸ After 3 years: M = 200 · (0.90)³ = 200 · 0.729 = 145.8 grams ## Percent Change and the Growth Factor The relationship between percent change and the growth factor is critical: | Situation | Growth Factor (b) | |---|---| | Increases by 15% per year | b = 1 + 0.15 = 1.15 | | Increases by 100% (doubles) | b = 1 + 1.00 = 2 | | Decreases by 20% per year | b = 1 − 0.20 = 0.80 | | Decreases by 50% per…