Assignment 2

Examining
the Parabola

By Cara
Haskins, Robin Kirkham, and Matt Tumlin

___________________________________________________________________

We will examine the parabola as the coefficient values change to see the effects these changes have on the various parabolas.

Examine the parabola
of y = ax^{2}+bx+c for different values of a, b, and c.

Our first step will be to:

Look at the basic parabola when a=1, b=0 and c=0

Notice that the** Domain** is the set of all real numbers, and the **Range** is all non-negative numbers. The range for this basic
parabola is all non-negative numbers.

The lowest point on a parabola is called the Minimum. The minimum point for our basic
parabola is (0,0).

**Step 1:** y=ax^{2}+bx+c

Let b=0,c=0, and vary the values of ÔaÕ. Our new equation becomes y=a
x^{2}.

Let us use the graphing calculator to examine the effects of using different values for a, remembering to use positive and negative values.

Notice that the minimum of the graphs does not change even
though the value for ÔaÕ was varied.
When ÔaÕ >1, the graph has been narrowed horizontally, resulting in a
horizontal shrinking of the graph. When 0 < a < 1, the graph has now been
stretched horizontally.

This leaves us with the question, what happens when
negative values are substituted for ÔaÕ?

By substituting negative values for **Ôa**Õ, notice there is a reflection across the x-axis for our
two graphs as well as a horizontal change of the basic parabola. The highest point on a parabola is
called a Maximum. The maximum on
our parabola is (0,0).

**Step 2:** Now we are
examining the effects of variable ÔbÕ.
Let a=1, and c=0 and change the values for b. Our new equation
is now y = x^{2}+bx.

Notice that the widths of the
parabolas remained the same, while the location of the minimum changed. This movement appears to be equal to
the value of Ðc/2 both vertically and horizontally. This is investigated in our step 3
below.

**Step 3: **Let us
again start with equation y=ax^{2}+bx+c. Let
a=1, b=0, and vary c, resulting in y = ax^{2}+c. Note this
investigation is not complete until we review the effects that variable ÔbÕ
might have with ÔcÕ.

The value of variable ÒcÓ moves the parabola up or down..
When c > 0, the graph moves to the left. When c < 0,the graph moves to
the right.

This vertical movement changes with respect to our minimum
point. This vertical shift appears
to be equal to the value of ÔcÕ.
To be sure, let us check that with a change to variable ÒbÓ
simultaneously.

This shows that the horizontal and vertical shifts are a
result of both the variables ÔbÕ and ÔcÕ.
The horizontal shift still appears to be Ðc/2 while the vertical shift
appears to be smaller than ÔcÕ. To
be sure, let us investigate further all three variables with respect to each
other.

** **

** **

This shows us that the horizontal and vertical shifts are a result of all a, b, and c respectively. The horizontal shift turns out to be Ô-bÕ/2a while a vertical shift turns out to be a

This set of parabolas introduces many interesting
effects. Firstly, one can see that
the y=ax2+bx+c, where a, b, and c, are all positive and the similar parabola
where ÔaÕ is the additive inverse, one observes that these two parabolas are
inverses and both shifted to opposite quadrants around the

** **

In summary, given the equation y = ax^{2} +bx+c)
the following are true:

Changes in the value of ÔaÕ effects the altitude of the
graph.

Changes in the value of ÔbÕ** **effects the of the graph.

Changes in the value of ÔcÕ effects in conjunction with ÔbÕ
together effect the phase shift of the graph.