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One thing you may have noticed is that there are many different parabolas that can be drawn through the two constant points. Changing the second differences affected the first differences and the other \((x, y)\) points in the table.
Because of this, changing the second differences changed the shape of the parabola as well as possibly the direction in which it opened.
- When the second differences are positive, the parabola opens up.
- When the second differences are negative, the parabola opens down.
Opens Up
Opens Down
You may also have noticed that sometimes the graph was a line.
This happens when second differences are \(0\), since in order for this to happen the first differences must all be equal.