Exponential Growth and Decay

Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to growth at an increasing rate. Example: Population growth under ideal conditions.

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. Example: Radioactive decay of a substance.

The formula is: $$y = a(1 + r)^t$$ where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

The formula is: $$y = a(1 - r)^t$$ where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

The y-intercept is the value of the function when x=0, which is equal to the initial amount 'a' in the formula.

Depreciation can be calculated using the formula: $$V = P(1 - r)^t$$ where 'V' is the value after depreciation, 'P' is the initial value, 'r' is the depreciation rate, and 't' is time.

A common example is compound interest in finance, where the amount of interest earned grows exponentially over time.

A common example is the decay of a radioactive substance, where the quantity decreases exponentially over time.

A negative growth rate indicates a decrease in quantity, characteristic of exponential decay.

Linear functions increase or decrease by a constant amount, while exponential functions increase or decrease by a constant percentage.