Since the function has an asymptote, it needs a constant of 60 000.
A
a
Exponential function
B
b
y=12 000(0.93)^x+60 000
Practice makes perfect
a
Notice the word level off. This suggests that the population approaches 60 000 over time. Therefore, this cannot be a linear relationship, since this would not show a population that levels off.
Instead we should use an exponential model.
b
From Part A, we know that the function illustrates exponential decay. Therefore, we can write it on the following format.
y=ab^x, b<1
However, the function has an asymptote at y=60 000. To reflect this fact, we must add 60 000 to the equation's right-hand side.
y=ab^x+60 000As x grows the value of ab^x decreases, which means the function approaches the constant.
Finding a and b
From the exercise, we know that the function's initial value is 72 000. We also know that two years after 1998, the population has dropped to 70 379. With this information, we can identify two datapoints and substitute them into the function.
rcl
( 0, 72 000)& → & 72 000= ab^0+60 000
( 2, 70 379)& → & 70 379= ab^2+60 000
Any power raised to 0 equals 1. With this information, we can solve for a in the first equation.
