Exponential growth and decay describe situations where a quantity changes by a constant percentage over equal time intervals. This pattern appears in population growth, compound interest, radioactive decay, medicine levels in the body, and cooling. Unlike linear change, the amount added or removed is not constant.
The curve bends because each new change depends on the current amount.
Key Facts
- General exponential form: y = a b^x, where a is the initial value and b is the growth or decay factor.
- Exponential growth occurs when b > 1, so y increases as x increases.
- Exponential decay occurs when 0 < b < 1, so y decreases as x increases.
- Percent growth model: y = a(1 + r)^t, where r is the growth rate as a decimal.
- Percent decay model: y = a(1 - r)^t, where r is the decay rate as a decimal.
- Continuous exponential model: y = a e^(kt), with k > 0 for growth and k < 0 for decay.
Vocabulary
- Exponential function
- A function in which the variable appears in the exponent, commonly written as y = a b^x.
- Initial value
- The starting amount of an exponential model, represented by a in y = a b^x.
- Growth factor
- The number multiplied each time the input increases by 1, with values greater than 1 producing growth.
- Half-life
- The time it takes for a decaying quantity to decrease to half of its current value.
- Doubling time
- The time it takes for a growing quantity to become twice as large.
Common Mistakes to Avoid
- Using the percent rate instead of its decimal form is wrong because 6% must be written as 0.06 in formulas like y = a(1 + r)^t.
- Treating exponential change like adding the same amount each step is wrong because exponential models multiply by the same factor each step.
- Confusing growth factor with growth rate is wrong because a 20% increase has rate 0.20 but factor 1.20.
- Using a decay factor greater than 1 is wrong because decay must have 0 < b < 1, such as 0.85 for a 15% decrease.
Practice Questions
- 1 A bank account starts with $800 and earns 5% interest per year. Write an exponential model and find the balance after 6 years.
- 2 A radioactive sample has 120 grams and a half-life of 4 hours. How much remains after 12 hours?
- 3 Two models are y = 50(1.08)^t and y = 50(0.92)^t. Explain which represents growth, which represents decay, and how the graphs would differ over time.