Question

1) City A and City B have increased growth in population. Write an exponential function to represent the City B’s population, y, based on the number of years that pass, x, after a period of exponential growth.

2) City A and City B have decreased growth in population. Write an exponential function to represent City B’s population, y, based on the number of years that pass, x, after a period of exponential decline.

3) Explain the similarities and differences between your GROWTH equations and your DECAY equations.

Answer 1

1) An exponential function to represent City B's population with increased growth can be written as:
y = A * b^x

where A is the population at time x = 0, b is the growth factor, and x is the number of years that have passed. The growth factor, b, is greater than 1.

2) An exponential function to represent City B's population with decreased growth can be written as:
y = A * b^(-x)

where A is the population at time x = 0, b is the decay factor, and x is the number of years that have passed. The decay factor, b, is between 0 and 1.

3) The similarities between the growth and decay exponential functions are that in both equations, the population is a function of time (x) and the exponential function is used to model the change in population over time.
The differences between the growth and decay exponential functions are that the growth function has a growth factor (b) greater than 1, while the decay function has a decay factor (b) between 0 and 1. Additionally, in the growth function, the exponent is positive (x), meaning the population is increasing, while in the decay function, the exponent is negative (-x), meaning the population is decreasing.