In order to determine whether a table of x-y values forms a quadratic relationship between the x and y variables or not, get the second differences between the given set of y-values. If the second differences between each consecutive pair of y-values is equal, then the given table of x-y values forms a quadratic relationship between the x and y variables.

For example, in the following table of x-y values, the second differences between each pair of y-values are constant for all y-values:

The second difference of each pair of corresponding y-values is 8. Since the second difference is same for all corresponding y-values of the table, therefore it represents a quadratic relationship between x and y. The quadratic equation for the above table is \[y={x}^{2}+2\,x+3\]

On the other hand, if a table of x-y values does not represent a quadratic relationship, the values in the second difference column will differ.

For example, in the following table of x-y values, the second differences between each pair of y-values are constant for all y-values:

X | Y | First difference, F_{n} = Y_{n} - Y_{n-1} | Second difference, S_{n} = F_{n} - F_{n-1} |
---|---|---|---|

2 | 11 | ||

4 | 27 | 27 - 11 = 16 | |

6 | 51 | 51 - 27 = 24 | 24 - 16 = 8 |

8 | 83 | 83 - 51 = 32 | 32 - 24 = 8 |

The second difference of each pair of corresponding y-values is 8. Since the second difference is same for all corresponding y-values of the table, therefore it represents a quadratic relationship between x and y. The quadratic equation for the above table is \[y={x}^{2}+2\,x+3\]

On the other hand, if a table of x-y values does not represent a quadratic relationship, the values in the second difference column will differ.

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