Consider the table for the quadratic function y=x^2+3x+12.

x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 |

y | 22 | 16 | 12 | 10 | 10 | 12 | 16 | 22 |

1st difference | 6 | 4 | 2 | 0 | -2 | -4 | -6 |

you can see in the first difference there is a pattern coming as 0,positive and negative 2,4,6…

Lets see an another example. consider y=x^2-5x+12

x | -1 | 0 | 1 | 2 | 3 | 4 | 5 |

y | 18 | 12 | 8 | 6 | 6 | 8 | 12 |

1st difference | 6 | 4 | 2 | 0 | -2 | -4 |

Here also the first difference follows the same pattern.

The general form of a quadratic function is y=ax^2+bx+c.This pattern will come if a=1 and b is an odd number.

## Lets visualise this pattern graphically

Graphically you can see this pattern now.If we plot the points for the above second example, the points follow the pattern 0-2-4-6… since the x and y co-ordiantes of the vertex are decimals it will not be in the calculation table and hence in the graph.Now you can understand -2,-4,-6 indicates they will be in the opposite direction.

- If a=1
- b is an odd number

Then the pattern will be even numbers as 0,2,4,6…

You can check with some more examples.

.