Varying ÔbÕ in the Standard Quadratic Equation
By Sharon K. OÕKelley
I. Consider the quadratic equation in standard formÉ
LetÕs examine graphs of a quadratic equation in which
ÒaÓ and ÒcÓ are held constant and ÒbÓ is varied.
In the figure below, the beginning equation isÉ
and ÒbÓ is varied.
From the graphs, you can draw
several conclusionsÉ.
1. When ÒbÓ is 3 and 3, the
parabolas intercept the xaxis in two places each. In other words, each
parabola has two zeroÕs or
two real solutions. These can be found by
setting the equation equal to 0 and solving for x. For exampleÉ.
(Solved using the
Quadratic Formula.)
2. When ÒbÓ is 2 and 2, the
parabolas touch the xaxis in one place each. In other words, each parabola has
a double zero or
one real solution. It is called a double zero in
that each equation can be factored into identical parts. For exampleÉ.
3. When ÒbÓ is 1 and 1, the
parabolas do not touch nor intersect the xaxis; therefore, there are no zeros
and hence no real
solutions as demonstrated belowÉ.
4. When ÒbÓ is 0, the graph does
not touch nor intercept the xaxis; therefore, there is no real number solution
to the
corresponding equation. In addition, when ÒbÓ is
0 the equation becomesÉ
When viewed in vertex form, the equation becomesÉ
From the above equation, the vertex can easily be identified
as (0, 1).
II. Next, letÕs find the vertices of each of the graphs using
the vertex formula for quadratic equations in standard formÉ.
The work will be given for the first
problem onlyÉ.
* If
b = 3É.
Therefore, the vertex is
(1.5, 1.25).
Summary:
Equation 
Vertex 

(1.5, 1.25) 

(1.5, 1.25) 

(1, 0) 

(1, 0) 

(0.5, 0.75) 

(0.5, 0.75) 

(0, 1) 
III. Next, considerÉ
graphed in purple with the other graphs below.
A hypothesis suggested by the figure
is that the graph of the purple parabola contains the vertices of the other
parabolas.
This can be verified with a table of
values forÉ
.
x 
y 
1.5 
1.25 
1 
0 
0.5 
0.75 
0 
1 
0.5 
0.75 
1 
0 
1.5 
1.25 
Also, notice that the purple
parabola is a reflection of the graph whose ÒbÓ is 0.
Thus, it can be stated that
the reflection of a graph of a quadratic equation in which ÒbÓ is 0 will
contain the vertices of
the parabolas created when ÒbÓ
is varied and ÒaÓ and ÒcÓ are held constant in the original equation.
Example:
If the hypotheses is true, thenÉ
should be the reflection ofÉ
and should contain the vertices of the parabolas created
when only ÒbÓ is varied in the original equation.
Red: b = 3, 3
Blue: b = 2, 2
Green: b = 1, 1