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