Gregory Schmidt

Write-up #2

Parabolic Investigations

Problem: Produce 5 to 10 graphs of

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

We will begin by considering our graph with .  So let  and

.

(1)

Now we notice that the two graphs both have the same y-intercept of .  But we need not graph this in order to find the y-intercept of a graph of the form .  Setting , we see that our graph will always have a y-intercept of .  When we solve for the x-intercepts of  and , we see that they are  and , respectively.  Hence, the x-intercepts of our graphs are symmetric about the y-axis.

Finally, we see that  and ,  both have a vertex of  and .  As we will see, all graphs of the form  will have a vertex of .

Let us now consider the graphs with , so that we have

and .

(2)

Again, we can graphically verify what we already now, and that is that our y-intercepts are .  These graphs have the same shape as the previous two, and only differ by the x-intercept and y-intercept.  As previously mentioned, we can see that  and  have a vertex of  and , respectively.

Graphing all of these at once we have:

(3)

All of these graphs are of the same class of graphs, which is why they all appear so similar.  As we have seen, varying , merely changes the x-intercept, the y-intercept, and the vertex, but leaves the overall shape of each graph unchanged.

This leads us naturally to question the shape and position of graphs of the form

.

What conclusions can one make based on the discoveries from the previous graphs?

Well, graphs of this form are certainly quadratic, and so should have the same general shape as the previous graphs.  Furthermore, setting , in order to determine the y-intercept gives us , which while different than before, is still quite similar in its relation to  and .  Finally, we see that graphs of this form have a vertex of .

Let us consider an example.  Take .  So we have

and .

(4)

If we desired to reflect these graphs about the y-axis, we now see that we can do so by considering graphs of the form .

Again let us consider for .  So we have:

(5)

Notice there is no difference in the shape of these graphs.

(6)

In fact, all of the graphs we have consider thus far have the same shape, and differ only with respect to their intercepts on the xy-axis, as well as their vertices.

Notice:

All of the graphs we have seen here of the same class of graphs, namely quadratic forms.