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.
Understanding Math: Exponential Growth and Decay
Doubling time is the time needed for an amount to become twice as large. It is useful because it describes the speed of growth in a way people can picture. A bacteria culture that doubles every hour goes from one hundred cells to two hundred, then four hundred, then eight hundred.
The same time interval produces bigger numerical increases later because the starting amount is larger each time. This helps explain why early growth can look slow on a graph, then become steep very quickly.
A fixed doubling time does not mean the same number is added each hour. It means the amount is multiplied by two each hour.
Half-life works in the opposite direction. It is the time needed for half of a quantity to remain. If a radioactive sample has a half-life of five years, one hundred grams becomes fifty grams after five years.
After another five years, it becomes twenty-five grams. The amount never reaches exactly zero in the mathematical model. It keeps being halved, leaving smaller and smaller amounts.
Scientists use half-lives to estimate the age of rocks and fossils. Doctors use them when planning medicine doses, since a drug may leave the bloodstream at a rate linked to its half-life.
Doubling and half-life can be converted into a model for any time interval. Suppose an investment doubles every ten years. After one ten-year period, the factor is two.
For a single year, the factor must be a smaller number which, multiplied by itself ten times, gives two. Calculators can find this value. The same idea applies to half-life.
If an amount halves every eight days, the daily factor is the number which gives one half after eight repeated multiplications. This is why exponential models can handle times such as three days or twelve and a half years, not only complete doubling or halving periods.
Real data rarely follows one exponential rule forever. A population may grow rapidly while food, space, and water are plentiful. Later, limited resources slow the growth.
Money in an account may compound predictably, but deposits, withdrawals, fees, and changing interest rates alter the pattern. Cooling is often approximately exponential only when the surrounding temperature stays steady. When solving problems, first identify the starting amount and the length of one time interval.
Then check whether the stated percentage, doubling time, or half-life matches the factor you use. Keep units consistent. A half-life in hours cannot be used directly with a time measured in days without converting one of them.
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.