What does it mean when the first differences of the y-values in the data set are almost equal?
Model: Linear Equation: y=- 0.5x+2
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.
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 take a look at our table.
x
y
0
2
1
1.52
2
1
3
0.49
4
0
We can exclude the possibility of an exponential model, since the y-value in the last pair is 0. Let's check for a common difference!
x
y
First differences
0
2
1
1.52
-0.48 ↩
2
1
-0.52 ↩
3
0.49
-0.51 ↩
4
0
-0.49 ↩
The first differences of the y-values are all approximately - 0.5, so a linear model fits the data.
Writing an Equation
We know that a linear function best models the given data.
y=mx+ b
To write an equation to model the data we have to determine the values of m and b. Recall that (0, b) is the y-intercept of the linear function. Since the pair (0,2) is included in the data set, b= 2.
y=mx+ 2
In order to determine the value of m we will use the Slope Formula and the points ( 0,2) and ( 1,1.52).
We have that m≈- 0.5. Note that the value of m is roughly the common difference! We could have used that information instead of the Slope Formula. Let's finish writing the equation to model the data!
y=- 0.5x+ 2
Below we have included a graph that shows how the equation models the given data.
