Exponential Growth vs Decay Comparison
Compare exponential growth and decay side by side. Adjust the initial value and base to see how doubling time, half-life, and growth rate change. Includes real-world presets like compound interest and radioactive decay.
CONCEPT: CONSTANT RATIO CHANGE
Exponential functions represent situations where a quantity changes by a constant percentage (ratio) over equal intervals. This differs from linear functions, which change by a constant amount. An exponential function grows if the base and decays if .
MECHANISM: BASE AND GROWTH RATE
The base is related to the growth/decay rate by . For growth, (e.g., 5% growth means ). For decay, (e.g., 5% decay means ). The initial value determines the y-intercept, as .
HOW TO USE THIS VISUALIZATION
1. **Adjust the Base ()**: Observe how the curve steepens as increases from 1, and how it flips to decay as drops below 1. 2. **Compare with Linear**: Enable the linear function to see that exponential growth always eventually exceeds any linear growth, no matter the starting values. 3. **Trace Points**: Move the cursor along the curve to see the constant ratio between successive y-values for equal steps in x.
CORE FORMULAS
AP EXAM CONNECTION
Unit: Unit 2: Exponential and Logarithmic Functions (Topic 2.1)
Learning Objective: LO 2.1.A
COMMON MISCONCEPTIONS
- Thinking a base of 1 leads to growth (it is constant).
- Confusing the base with the growth rate .
- Believing exponential functions eventually turn back down.
KEY TAKEAWAYS
- Growth: ; Decay: .
- Exponential functions have a constant multiplier, not a constant slope.
- Exponential growth eventually dominates any polynomial growth.
PRACTICE QUESTIONS
Q1 (QUANTITATIVE): If a value decreases by 12% each year, what is the base of the exponential function?
Show Answer & Explanation
Answer: 0.88
Explanation: . Since it is decreasing, . Thus, .
