Consider the table for the quadratic function y=x^2-2x+8.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 23 | 16 | 11 | 8 | 7 | 8 | 11 | 16 |
| 1st difference | 7 | 5 | 3 | 1 | -1 | -3 | -5 |
you can see in the first difference there is a pattern coming as positive and negative 1,3,5,7…
Lets see an another example. consider y=x^2+4x+12
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 |
| y | 17 | 12 | 9 | 8 | 9 | 12 | 17 |
| 1st difference | 5 | 3 | 1 | -1 | -3 | -5 |
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 even number(including 0)
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 1-3-5… from the vertex.Now you can understand -1,-3,-5 indicates they will be in the opposite direction.
- If a=1
- b is an even number
Then the pattern will be odd numbers 1,3,5…
You can check with some more examples.