### Assignment 2

*Explorations of Second Degree Equations*

### Marianne Parsons

Using Graphing Calculator 3.2, we are able
to explore the results of:

for different values of a.

This second degree equation can be shown when
a = 1
as:

What can we expect...

**for intersections of all graphs?**

Because of the + 2 on this end of this equation,
our parabola will always intersect the y-axis at y=2. No matter
what value is chosen for a, we can see when x=0, y=2. Therefore,
in our exploration of this equation we can expect that all of
our parabolas generated will produce graphs that intersect at
the common point (0,2). To see the basic algebra steps, click
here.

**when a = 0?**

We can anticipate what the graph would like
like when a=0. Using basic algebra skills, when a=0, our resulting
equation is no longer of the second degree. This means, our resulting
equation will not be a parabola, but will instead be a line with
the equation y=x+2. To see the basic algebra steps, click here. While exploring this equation
we can already see a line will be generated instead of a parabola,
only when the equation is no longer of the second degree, or
when a=0.

**when a > 0?**

The parabola generated when a is a positive
value will open upward. This means all corresponding y values
on the parabola will be greater than the y value expressed at
the vertex. For a diagrammatic explanation of positive a values,
click here.

**when a < 0?**

The parabola generated when a is a negative
value will open downward. This means all corresponding y values
on the parabola will be less than the y value expressed at the
vertex. For a diagrammatic explanation of negative a values,
click here.

Testing our expectations, we can graph the
following equations for varying values of the coefficient a:

As a approaches zero the corresponding parabola widens,
and gets closer to the straight line when a = 0. As **a** gets larger,
the corresponding parabola becomes more narrow as shown above.
What do you think the graph would look like if the value of a
= 1/100? What about a = 100?

View the animation of our parabola as the value
of **a** goes from -10 to 10.

Return
to Class Page