Check the ratios of the numbers of frogs. What do your results mean?
Model: Exponential Equation: y=120(0.8)^x
Practice makes perfect
We will start by determining the type of function that best models the data. Then we will be able to write an exact equation that models the data.
Finding a Model
We want to find the most appropriate model for the given data set. Note that the years have a common difference of 1. Therefore, we can check if the numbers of frogs have a common difference, a common ratio, or constant second differences. It will tell us which model is most appropriate for the data set.
Sometimes data does not fall exactly into the models listed above. In this case we have to round our results and see when they are the closest to being constant. Let's start by checking our data set for a common difference!
Year
Number of Frogs
First Differences
0
120
1
101
-19 ↩
2
86
-15 ↩
3
72
-14 ↩
4
60
-12 ↩
As we can see, the differences between the consecutive numbers of frogs are far from being constant. Therefore, we will check for a common ratio next.
Year
Number of Frogs
Ratio
0
120
1
101
101/120=0.841...
2
86
86/101=0.851...
3
72
72/86=0.837...
4
60
60/72=0.833...
The ratios between the consecutive values of the balance are all roughly 0.8, so an exponential model fits the data.
Writing an Equation
We know that an exponential function best models the number of frogs.
y=a*b^x
To write an equation to model the data we have to determine the values of a and b. Recall that (0,a) is the y-intercept of the exponential function. Since the pair (0,120) is included in the data set, a=120.
y=120*b^x
In order to determine the value of b we will use one (x,y) pair from the data set, other than (0,120), to write an equation. Let's use ( 1, 101).
We have that b≈0.8. Note that the value of b is equal to the approximation of the common ratio! We could have used that information instead of solving the equation. Let's finish writing the equation to model the data!
y=120*0.8^x ⇔ y=120(0.8)^x
