Which equation models exponential decay?

Which equation models exponential decay?

• A. y = a(1+r)^t
• B. y = a(1-r)^t
• C. y = a r^t
• D. y = a - rt

Answer: (B)

Exponential behavior changes by multiplying by a constant factor each time unit. If that factor is between 0 and 1, the quantity decays over time. In the equation y = a(1 − r)^t, the base is (1 − r). When r is a positive decay rate, this base is a number between 0 and 1, so each step multiplies the current amount by a fraction, causing the value to shrink as t increases. The other forms behave differently: y = a(1 + r)^t grows when r > 0 because the base is greater than 1; y = ar^t is exponential with base r, which decays only if 0 < r < 1 but isn’t phrased to emphasize a constant fractional decrease per time unit; and y = a − rt is linear, not exponential.

Exponential behavior changes by multiplying by a constant factor each time unit. If that factor is between 0 and 1, the quantity decays over time. In the equation y = a(1 − r)^t, the base is (1 − r). When r is a positive decay rate, this base is a number between 0 and 1, so each step multiplies the current amount by a fraction, causing the value to shrink as t increases.

The other forms behave differently: y = a(1 + r)^t grows when r > 0 because the base is greater than 1; y = ar^t is exponential with base r, which decays only if 0 < r < 1 but isn’t phrased to emphasize a constant fractional decrease per time unit; and y = a − rt is linear, not exponential.