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?

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.

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) ?