We will begin with the most basic parabola with

a=1

b=0

c=0.

Notice that the vertex is at (0,0).

What does "a" do?Look at a few graphs of equations that vary the value of

ain the equationa= 1,-1, 2, 3, .5, 10, -.05

We immediately realize that

aaffects thewidthof the parabola as well as thedirectionthat the the parabola opens (up or down). Ifais negative, the the parabola opens downward. Ifais positive, it opens upward.As

agets larger, the parabola gets skinnier. Asagets smaller, the the parabola gets wider.What does "c" do?Next, look at some graphs of equations where

a=1, b=0, and the value ofcvaries:c= 0, 1, -1, 4, -4

It appears that **c** affects the **vertical
location** of the vertex. When **c** is **POSITIVE**,
the original vertex moves **UP** **c** units. When **c**
is **NEGATIVE**, the vertex moves **DOWN c **units.

Now let's look at parabolas where **a=1, c=0,
**and **b** varies.

We notice that **b** also affects the location
of the vertex. Recall that **c** affected the vertical location
of the vertex. As the value of **b** varies from **-4 to 4**,
there is a shift in the location of the vertex. When **b**
is negative, there is a shift to the right. When **b** is positive,
the shift occurs to the left. Also notice that there is a downward
shift from the original parabola in both instances. Therefore,
**b **affects both the **horizontal** and **vertical**
positioning of the vertex.

It would be interesting to look at a continuous
range of values of **b**.

**But how do we know exactly where the vertex
is at all times?**

First, we let **b=0 **and **c=0** while
we varied **a. **We discovered that **a **affects the width
and direction of the parabola.

Then we let **a=1 **and **b=0 **while
we varied **c.** We discovered that **c** affects the vertical
position of the vertex of the parabola

Then we let **a=1** and **c=0** while
we varied **b.** We discovered that **b **affects the horizontal
and vertical position of the parabola.

Let's look at a few more graphs and then we'll
try to predict where the vertex is. Here the results are a little
more interesting because we let **b=1** and **c=1** while
we vary **a.** This is different because we are no longer looking
at a parabola whose vertex is at **(0,0). **At first when we
varied **a **while **b=c=0**, the vertex never changed positions;
the width and direction of the parabola were the only changes.
Now notice that the vertex also changes its location as we vary
**a. **Notice that when **a=0**, you get the graph of a
line that is tangent to all of the parabolas whose **b** and
**c** are kept constant.

This indicates that the value of **a** also
affects the location of the vertex.

I personally did figure this out by trial and error!!! (HONESTLY). But then a friend, Nicole Mosteller, helped me justify this with a little more sophistication.

If you think of all the lines that are tangent to the parabola in the above form, you realize that there is one line whose slope is zero. The point of tangency for this particular line is the vertex. If we take the derivative of the quadratic equation above, we get

Recall that **y'** is the slope of the line
tangent to the graph at a given point. We are interested in knowing
what the cooordinates are when the slope, y', is equal to zero.

So we do a little bit of algebra and solve
for **x **and **y**, the coordinates of the vertex:

Now, solve for **y **by substituting for **x** in the
original equation**:**

So, we have the coordinates of the vertex.