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 x-axis 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 x-axis 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 x-axis; 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 x-axis; 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

 

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