Exploring parabolas:

y = *a*x² + *b*x +*c*

By: Lauren Lee

The
above equation is known as a quadratic equation. A **quadratic equation**
is an equation that can be expressed in the form of:

where
** a** is not equal to zero.

Every
quadratic equation has U-shaped graph called a parabola.

For my first
exploration, I want to start out by looking at a parabola with a fixed *a*
and *b* constant. Let *a*=1, *b*=1,
and vary *c* values.
Thus our equation is: y = x² + x
+ *c*.

Let’s look at the graph for values *c* = -1,
*c*
= -4, *c* = 0, *c* = 1, *c* = 5.

** **

What is seen in our picture is that as the values of ** c
**increase, the parabola narrows and the vertex increases. Similarly, as

Now
let’s explore what happens when we fix ** a** and

**Let’s look at the graph for values ***b*** = 0****, ***b*** = -2****, ***b*** = -4****, ***b*** = -7****, ***b*** = 2****, ***b*** = 4****, b = 7.**

** **

We
can see that the 'highest' parabola is x² + 0x + 1,
which also happens to be the upper limit for the parabolas. As ** b** increases positively, the
parabola dips lower and lower to the left. As

Finally,
let’s explore what happens when we fix ** b** and

**Let’s look at the graph for values ***a*** = -1****, ***a ***= -2****, ***a ***= -4****, ***a*** = 0****, a = ½, a = 1, a = 2, a = 4. **

We initially notice that when ** a** is positive,
the parabola opens upward (concave up).
Similarly, when

Finally, we notice that as **| a|** increases,
the parabola narrows, and as

Do some more explorations on your own. How does the shape change? How does the position change?