by Oktay Mercimek

Problem: Produce several ( 5 to 10) graphs of 

on the same axes using different values for d. Does varying d change the shape of the graph? the position?

Let's start with 0 (zero)


For d=1 it is,




It seems there is not much change except graph is sliding. I also seems like vertex is related both d and -2.

In the first graph, vertex is on the point (0,-2) where d=0.

In the second graph, vertex is on the point (1,-2) where d=1.

 Let's try a few other values and see whether our assumption is true or not.


Click Here for Graphing Calculator File

There is not much change on the lowest points (vertexes) of parabolas. All these points have -2 as y coordinate and d as x coordinate.

Let's look on this issue using mathematical reasoning.

What we look on this parabola   is vertex is at point (d,-2)

  then    so , and .

Then we found .

Since Δ=8 then there are two roots for this equation and Δ is independent from d.

these roots are in the form of  .

In parabola, we know  is in the middle of two roots.

Then vertex's x coordinate is .

So the vertex's x coordinate in our parabola is

That means our assumption about vertex's x coordinate is true.

Now look at the y coordinate.

If the vertex's x coordinate is d , then 

and y=-2. So vertex is always at the point (d,-2)

Can we assume vertex of  is always at the point (d,k) ?