Use patterns between consecutive data pairs to determine what type of function models the data.
See solution.
Practice makes perfect
Given a set of data, we can use patterns between consecutive data pairs to determine what type of function best models the data. The differences between consecutive y-values are called first differences. The differences between consecutive first differences are called second differences.
Linear Function: The first differences are constant.
Quadratic Function: The second differences are constant.
Exponential Function: Consecutive y-values have a common ratio.
Remember that in all cases the differences of consecutive x-values need to be constant! Now, we will check if the method works for our data sets.
Linear Function
Data from the following table can be modeled by a linear function in the form y=2x.
Let's compute the first differences.
Since the first differences are constant, the data can be modeled by a linear function. The method works! âś“
Since the first differences are not constant, the data cannot be modeled by a linear function. Next, we will check the second differences.
Since the second differences are constant, the data can be modeled by a quadratic function. The method still works! âś“
Exponential Function
Finally, we will analyze data that models an exponential function. Let's choose y=2^x.
Just like before, let's compute the first differences.
Since the first differences are not constant, the data cannot be modeled by a linear function. Next, we will check the second differences.
Since the second differences are not constant, the data cannot be modeled by a quadratic function. Finally, we will check if the consecutive y-values have a common ratio.
Since the consecutive y-values have a common ratio, the data can be modeled by an exponential function. The method worked for all the examples! âś“
