To begin our exploration of second degree equations, lets first consider the general equation which can be written in the form:

The graph of the equation has the following
form. The graph of a second degree equation or *quadratic equation
*is a special type of U-shaped curve called a ** parabola**. These types of curves occur in many real life situations
or applications---especially situations involving reflective properties,
such as the inside of your eyes, satellite dishes, and flashlight
reflectors.

As a starting point, lets first take a look at the graph of

Notice that the bottom of the curve, called
the *vertex* is centered at the origin on the *Cartesian
coordinate *system. We can invert the graph by simply putting
a minus in front of the x-squared term. Here the parabola

Now returning to our original equation lets
see what happens when we "plug-in" values for **a**,
**b**, and **c**. We will start with the integer 2 by replacing
individually for a, then b, then c.

By replacing the **a** coefficient with
2 has the effect of narrowing the curve. By replacing the **b**
coefficient by 2 has the effect of shifting the whole curve to
the left by two units. Finally, by replacing **c** with 2 has
the effect of shifting the whole curve up by two units.

Now, let set b=1, c=2 and let a be 2, 1/4, 0, and -2.

In summary, the coefficient **a** has the
effect of either narrowing or widening the curve. The coefficient
**b** has the effect of shifting the curve horizontally, that
is, left of right. Finally, **c** has the effect of shifting
the curve vertically or up/down.