Exercise 44 Page 601

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.

x	y	First differences
- 1	11.2
0	7	-4.2 ↩
1	2.8	-4.2 ↩
2	- 1.4	-4.2 ↩
3	- 5.6	-4.2 ↩
4	- 9.8	-4.2 ↩

The first differences of the y-values are all - 4.2, 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 (0,7) is included in the data set, b=7. y=mx+7 In order to determine the value of m, we will use the Slope Formula and the points ( - 1,11.2) and ( 0,7).

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( - 1,11.2) & ( 0,7)

m=7- 11.2/0-( - 1)

▼

Subtract terms

SubNeg

a-(- b)=a+b

m=7-11.2/0+1

AddSubTerms

Add and subtract terms

m=- 4.2/1

DivByOne

a/1=a

m=- 4.2

We have that m=- 4.2. Note that the value of m is equal to 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=- 4.2x+7 Below we have included a graph that shows how the equation models the given data.