Question

what is the growth factor in an exponential function?
(1 + r) where r is the growth rate expressed as a decimal
(1 + 2r) where r is the growth rate expressed as a decimal
(1 + r^2) where r is the growth rate expressed as a decimal
(1 - r) where r is the growth rate expressed as a decimal

Explanation:

The general form of an exponential growth function is \( A(t)=A_0(1 + r)^t \), where \( A_0 \) is the initial amount, \( r \) is the growth rate (expressed as a decimal), and \( (1 + r) \) is the growth factor. Let's analyze each option:

Step 1: Analyze Option 1

Option 1: \( (1 + r) \) where \( r \) is the growth rate expressed as a decimal. This matches the definition of the growth factor in the exponential growth formula \( A(t)=A_0(1 + r)^t \).

Step 2: Analyze Option 2

Option 2: \( (1 + 2r) \) where \( r \) is the growth rate expressed as a decimal. There is no standard exponential growth model with a factor of \( (1 + 2r) \); the growth factor is typically \( (1 + r) \) for a single growth rate \( r \) per time period.

Step 3: Analyze Option 3

Option 3: \( (1 + r^2) \) where \( r \) is the growth rate expressed as a decimal. The growth factor in exponential growth is based on adding the growth rate to 1, not squaring the growth rate and adding to 1. So this does not represent a standard growth factor.

Step 4: Analyze Option 4

Option 4: \( (1 - r) \) where \( r \) is the growth rate expressed as a decimal. This would be a decay factor (for exponential decay, the formula is \( A(t)=A_0(1 - r)^t \) where \( r \) is the decay rate), not a growth factor.

Answer:

The correct option is the first one: \((1 + r)\) where \( r \) is the growth rate expressed as a decimal.