Question

how do you write an equation for exponential growth?
y = a(1 + r)^t, where a is the initial amount, r is the growth rate, and t is time.
y = a * r^t, where a is the initial amount, r is the growth factor, and t is time.
y = a + rt, where a is the initial amount, r is the rate of change, and t is time.
y = a(1 - r)^t, where a is the initial amount, r is the decay rate, and t is time.

Explanation:

To determine the correct equation for exponential growth, we analyze each option:

• The first option is \( y = a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth rate, and \( t \) is time. This matches the standard exponential growth formula, as exponential growth involves a base of \( (1 + \text{growth rate}) \) raised to the time exponent.
• The second option \( y = a * r^t \) does not represent growth correctly, as it lacks the \( (1 + r) \) structure (it would be a different kind of exponential model, not standard growth).
• The third option \( y = a + rt \) is linear (not exponential), as it has a constant rate of change added linearly.
• The fourth option \( y = a(1 - r)^t \) represents exponential decay (since it subtracts the rate), not growth.

Answer:

The correct equation for exponential growth is the first option: \( y = a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth rate, and \( t \) is time.