Which table represents a quadratic relationship?

A. x - 2 - 1 0 1 2 3

f (x) 4 2 1 0.5 0.25 0.125

B. x - 7 - 6 - 5 - 4 - 3 - 2

f (x) 135 128 105 72 35 0

C. x - 2 - 1 0 1 2 3

f (x) - 23.4 - 23.2 - 23 - 22.8 - 22.6 - 22.4

D. x - 1 0 1 2 3 4

f (x) 90 56 26 0 - 22 - 40

In each case, the x-values are equally-spaced. Thus looking at second differences will tell you if the relation is quadratic. If the second differences are non-zero and constant, then the values have a quadratic relationship.

A. First differences are 2-4 = - 2, 1-2 = - 1, 0.5-1 = - 0.5. Second differences are - 1 - (-2) = 1, - 0.5 - (-1) = 0.5. Since 1 ≠ 0.5, this relation is not quadratic. (It is exponential with a base of 1/2.)

B. First differences are 128-135 = - 7, 105-128 = - 23, 72-105 = - 33. Second differences are - 23 - (-7) = - 16, - 33 - (-23) = - 10. Since - 16 ≠ - 10, this relation is not quadratic. (It is cubic, since 3rd differences are constant at + 4.)

C. First differences are - 23.2 - (-23.4) = 0.2, - 23.0 - (-23.2) = 0.2, - 22.8 - (-23.0) = 0.2. Second differences are zero, so this is not a quadratic relation. (It is linear, with a slope of 0.2.)

D. First differences are 56-90 = - 34, 26-56 = - 30, 0-26 = - 26. Second differences are - 30 - (-34) = 4, - 26 - (-30) = 4. These are constant (=4), so the relation is quadratic.

The appropriate choice is ...

... D. x - 1 0 1 2 3 4

... f (x) 90 56 26 0 - 22 - 40