The Problem: Interpret your graphs. What happens to

This is the case where b=1
and c=2 as **a** is varied.

Is there a common point to
all of the graphs? What is it? What is the significance where
**a** =0?

Do similar interpretations for other sets of graphs. How does the shape change? How does the position change?

First Let's look at a a few different graphs
with different values of **a.**

With these three graphs we notice that a large
value of **a** seems to make the bowl-shape of the parabola
narrower. The smaller value of **a** makes the bowl-shape wider.
The vertex, or the minimum value, moves, yet the three parabolas
all include the point (0, 2).

Now let's see what happens when **a** is
negative.

With a negative **a**, we notice that the
parabola has flipped over, but still the graph includes the point
(0,2).

Now let's look at the case where **a**=0.

Since a =0, the term with the exponent = 2 ceases to exists. This gives us the equation y=x+2, which is not a quadratic, this finction is linear, meaning it graphs a line, as the graph above shows. We should still note that the point (0,2) is on the graph of the line.

Let's take a look at all of the graphs as **a**
varies.

Click **here**
to open a graph in graphing calculator 3.1

The next case that we need to look at is when
**a** and **b** are fixed and **c** varies.

We can see here that the bowl-shape of the graph remains the same, but the graph is shifted up or down based on the value of c.

Now, let's look at the interesting case where
**b** varies.

Varying values of **b** changes the location
of the graph significantly, without changing the shape of the
graph. Notice that all of the graphs go through the point (0,-2)
which is implied by c=-2.

The vertices of the parabolas appear to be plotted along a parabolic curve. They appear to peak at the point (0,-2).

Click **here**
to see a graph in graphing calculator 3.1

The equation for a parabola may be written in the form

where h and k are x and y values of the vertex,
(h,k). So, in our example, h=0, and k=-2. We know that the parabola
will open DOWN, so our value for **a** must be negative.

The points making this curve are the vertices from a family of parabolas with a=2.; it follows that the two are related. So, now we try:

Click **here**
to see a graph in graphing calculator 3.1

Notice in the graph that the vertices of the verying parabolas follow the path with the eqation above.

Conjecture:

The vertices of the graph of

as n varies,

will follow the parabolic path described by the equation