The Graph of

EMAT 6680 - Assignment 2

by Mary Bruce

In this assignment the author examines multiple graphs of the form     for different values of d.

The simplest form to look at would seem to be d=0. Thus, the graph of the equation    reduced to is as follows:

One notices the vertex of the parabola falls at the point (0, -2).

Next, choosing positive values of d (d=1,2,3,4) and producing graphs in sequential order to facilitate analysis of movement and change, one notices a horizontal translation of the graph d units to the right of the original.

The vertex appears to have shifted to (1, -2)  but otherwise no changes in the shape of the original graph are detected.

Looking at the graph of d=2,

again no changes in the shape have resulted though the vertex has now shifted horizontally to the point (2, -2).  The pattern continues for the graphs of d=3 and d=4:

Next, choosing negative values for d produces similar results. Using d= -1, we see a parabola with the same shape as the original shifted one unit to the left:

Notice the vertex has now changed to (-1, -2).  Graphs using d=-2, -3, -4 are as follows and illustrate similar patterns as above:

Even choosing fractional values for d produces expected results.  Using d=1/2 and d= -1/2, we notices horizontal translations 1/2 units to the right and left respectively:

Analyzing the graphs and equations simultaneously, it is easy to conclude why this particular quadratic form is often referred to as VERTEX  form:

y = (x - 0)^2 -2          vertex (0, -2)

y = (x - 1)^2 -2          vertex (1, -2)

y = (x - 2)^2 -2          vertex (2, -2)

y = (x - -1)^2 -2         vertex (-1, -2)

y = (x - -2)^2 -2         vertex (-2, -2)

Taking a quadratic equation       one completes the square

y = x^2 - 16x +64 +62 -64 to obtain      which should produce a parabola with vertex (8, -2) that maintains the same shape as the original:

Click here to view a Nucalc animation of the discussed translation. You will need to hit the play button next to n at the bottom of the screen.