Question

How can you tell whether an exponential equation models growth or decay? Use the general form of an exponential expression to explain your answer

Answer 1

To determine if an exponential equation represents growth or decay, examine the base of the expression: a growth model has a base greater than 1, while a decay model has a base between 0 and 1. Exponential growth is illustrated by a J-shaped curve, whereas logistic growth follows an S-shaped curve.

Step-by-step explanation:

The general form of an exponential function is f(t) = a*b^t, where a is the initial amount, b is the base, and t is the time.

Growth is modeled when the base b is greater than 1. This signifies that the quantity is increasing over time. For example, with a base of 2, the sequence would be 2, 4, 8, 16, and so forth, representing that the population doubles at each time interval.

In contrast, decay is modeled when the base b is between 0 and 1. This indicates that the quantity is decreasing over time, such as in the case of radioactive decay or depreciation of assets.

Exponential growth is often represented by a 'J-shaped' curve, which depicts how a population may grow faster as the population becomes larger. On the other hand, logistic growth, which is more realistic in natural populations due to factors like limited resources, follows an 'S-shaped' curve where growth levels off at carrying capacity.

Answer 2

Step-by-step explanation:

The general form of an exponential equation for growth is

`y=(1+r)^x`

and for decay is

`y=(1-r)^x`

In general, if the number inside the parenthesis (the growth or decay rate) is greater than 1, it's a growth problem. If the number inside the parenthesis is greater than 0 but less than 1 (in other words a positive fraction), it's a decay problem.