In exponential decay, how is the function generally represented?

In exponential decay, the function is typically represented as ( y = a(1 - r)^t ). In this representation, ( a ) is the initial amount, ( r ) is the decay rate, and ( t ) represents time. This formula shows how the quantity decreases over time by a fixed percentage.

The term ( (1 - r) ) reflects that with each time increment, the quantity is multiplied by a factor less than one, indicating that the value is decreasing. The exponent ( t ) signifies the number of time intervals that have passed, compounding the decay effect. For example, if ( r ) is 0.1 (or 10% decay), ( 1 - r ) would equal 0.9, demonstrating a consistent reduction.

The other options presented do not accurately describe exponential decay. They either indicate an increase or lack the necessary structure to convey continuous decay. This distinction is vital for understanding how quantities diminish over time in various applications, such as radioactive decay, depreciation of assets, or population decline. Thus, the correct representation for exponential decay is indeed ( y = a(1 - r)^t ).