What is 1-3-5 pattern in a quadratic function?

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.