What does it mean when the y-values in the data set have a common ratio?
Model: Exponential Equation: y=4(2.5)^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 determine the most appropriate model for the given data set. Note that the x-values have a common difference of 1. Therefore, we can check if the y-values have a common difference, a common ratio, or constant second differences. It will tell us which model is most appropriate for the data set.
As we can see, the differences between the consecutive y-values are not constant. Therefore, we will check for a common ratio next.
x
y
Ratio
- 1
1.6
0
4
4/1.6=2.5
1
10
10/4=2.5
2
25
25/10=2.5
3
62.5
62.5/25=2.5
4
156.25
156.25/62.5=2.5
The ratios between the consecutive y-values are all 2.5, so an exponential model fits the data.
Writing an Equation
We know that an exponential function best models the given data.
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 (0,4) is included in the data set, a=4.
y=4*b^x
In order to determine the value of b, we will use one (x,y) pair from the data set, other than (0,4), to write an equation. Let's use ( 1, 10).
We have that b=2.5. Note that the value of b is equal to 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=4*2.5^x ⇔ y=4(2.5)^x
Below we have included a graph that shows how the equation models the given data.
