Question

how do you write an equation for exponential decay? y = a(1 - r)^{\frac{t}{t}} y = a - rt y = a(r)^{\frac{t}{t}} y = a(1 + r)^{\frac{t}{t}}

Explanation:

The general formula for exponential decay is \( y = a(1 - r)^{\frac{t}{n}} \) (or in the form with \( \frac{t}{n} \) as the exponent, here the options have \( \frac{t}{n} \) represented as \( \frac{t}{t} \)? Wait, actually, the standard exponential decay formula is \( y = a(1 - r)^t \) for continuous or discrete with period 1, but when compounded or with a rate per period, if the exponent is \( \frac{t}{n} \), the formula is \( y = a(1 - r)^{\frac{t}{n}} \). Among the options, \( y = a(1 - r)^{\frac{t}{t}} \) (but likely a typo and \( \frac{t}{n} \) where n is the number of periods, but the first option \( y = a(1 - r)^{\frac{t}{t}} \) (assuming the exponent is \( \frac{t}{n} \) and maybe a formatting issue) is the exponential decay formula. The other options: \( y = a - rt \) is linear, \( y = a(r)^{\frac{t}{t}} \) would be growth if r >1, \( y = a(1 + r)^{\frac{t}{t}} \) is exponential growth. So the correct one for decay is the first option with \( (1 - r) \) in the base.

Answer:

The correct equation for exponential decay among the given options is \( y = a(1 - r)^{\frac{t}{t}} \) (the first option in the list, assuming the exponent's variable is a typo and represents the time division for the decay rate application). So the answer is the first option: \( y = a(1 - r)^{\frac{t}{t}} \) (or as presented in the image, the top - most option with \( y = a(1 - r)^{\frac{t}{t}} \)).