What is 0-2-4-6… pattern in a quadratic function?

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

x-5-4-3-2-1012
y2216121010121622
1st difference 6420-2-4-6

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-1012345
y1812866812
1st difference 6420-2-4

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.

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