Question

Complete the table of first differences, second differences, and/or first difference ratios to classify therelation.

Answer 1

Given:

Required:

We need to find the first differences, second differences, and first difference ratios.

Step-by-step explanation:

Recall that the first difference is the difference between values of the dependent variable by subtracting the previous value from each.

`(1)/(9)-(1)/(27)=(1*3)/(9*3)-(1)/(27)=(3)/(27)-(1)/(27)=(2)/(27)`
`(1)/(3)-(1)/(9)=(1*3)/(3*3)-(1)/(9)=(3)/(9)-(1)/(9)=(2)/(9)`
`1-(1)/(3)=(1*3)/(1*3)-(1)/(3)=(3)/(3)-(1)/(3)=(2)/(3)`
`3-1=2`
`9-3=6`

Recall that the second difference is the difference of the first difference.

`(2)/(9)-(2)/(27)=(2*3)/(9*3)-(2)/(27)=(6)/(27)-(2)/(27)=(4)/(27)`
`(2)/(3)-(2)/(9)=(2*3)/(3*3)-(2)/(9)=(6)/(9)-(2)/(9)=(4)/(9)`
`2-(2)/(3)=(2*3)/(1*3)-(2)/(3)=(6)/(3)-(2)/(3)=(4)/(3)`
`6-2=4`

Divide the second number by the first number to find the ratio of the two numbers in the first difference.

`(2\/9)/(2\/27)=(2)/(9)*(27)/(2)=3`
`(2\/3)/(2\/9)=(2)/(3)*(9)/(2)=3`
`(2)/(2\/3)=2*(3)/(2)=3`
`(6)/(2)=3`

We get the ratio of the first difference is constant so the given function is an exponential function.

Final answer:

`An\text{ exponential function}`