Consider the table for the quadratic function y=x^2-2x+8.
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
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.