By: J. Matt Tumlin, Cara Haskins, & Robin Kirkham
for the parabola y = ax2 + bx + c for different values of a, b, and
c (a, b, c can be any rational numbers).
Our first step
will be to look at the basic parabola
when a=1, b=0
Notice that the
Domain is the set of real numbers, and the Range for this basic parabola is all
use the basic quadratic function as our frame of reference. Let us examine what
happens to the graph under the following guidelines.
Step 1: Let b=0,c=0, and vary the values of a.
Our new equation becomes
Let us use
Graphing Calculator 3.2 to examine the effects of using different values for a,
remembering to use positive and negative values.
The red graph
is y= x2. This basic parabola will always be in red in future
examples for comparison purposes.
the minimum of the graphs does not change even though the value for ÒaÓ was
varied. When ÒaÓ > 1, the graph
has been narrowed horizontally, resulting in a horizontal shrinking of the
graph. When 0 < ÒaÓ < 1, the
graph has now been stretched horizontally.
the question what happens when negative values are substituted for variable
substituting negative values for ÒaÓ, notice there is a reflection across the
x-axis for our two graphs as well as a horizontal change of the basic parabola.
point on a parabola is called the maximum. The maximum point on our two
parabolas is (0,0).
Step 2: Now we
are examining the effects of variable ÒbÓ. Let a=1, and c=0 and change the values for b. Our new equation is now:
y = x2
the widths of the parabolas stayed the same, while the location of the minimum
changed. This movement appears to
be equal to the value of Ò-bÓ/2, both vertically and horizontally. This is investigated in our Step 3
Step 3: Let us start again with our original equation y=
ax2 + bx + c. Let a=1, b=0, and vary c, resulting in:
y = x2 + c
The value of
variable ÒcÓ moves the parabola up or down. When ÒcÓ > 0, the graph moves to
up. When ÒcÓ < 0, the graph
movement changes in respect to our minimum point. The vertical shift appears to be equal to the value of ÒcÓ.
To be sure, let us check what happens to a change of variable ÒbÓ
and ÒcÓ simultaneously.
This shows us the
horizontal and vertical shifts are a result of both the variables ÒbÓ and ÒcÓ
respectively. The horizontal shift
still appears to be Ò-cÓ/2, while the vertical shift appears to be smaller than
To be sure,
let us investigate what happens to a change of variable ÒaÓ ,ÒbÓ and ÒcÓ
This shows us
the horizontal and vertical shifts are a result of all the variables ÒaÓ, ÒbÓ
and ÒcÓ respectively. The
horizontal shift turns out to be ÒÐb/2aÓ, while the vertical shift turns out to
be the y value when x is equal to Ðb/2a.
This set of parabolas introduces many
introduces many interesting effects.
Firstly, one can see y= ax2 + bx + c where ÒaÓ, ÒbÓ, and ÒcÓ
are all positive and the similar parabola where ÒaÓ is the additive
inverse. One observes that these
two parabolas are inverses and both shift to opposite quadrants.
given the equation y = ax2 + bx + c the following are true:
á Changes in the
value of ÒaÓ effects the direction and width of the parabola.
á Changes in the value
of ÒbÓ effects the horizontal and vertical shift.
á Changes in the
value of ÒcÓ effects the vertical shift.
á Changes in the
value of ÒaÓ, ÒbÓ and ÒcÓ together effect the total shift of the parabola.
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