Varying ‘b’ in the Standard Quadratic Equation

By Sharon K. O’Kelley


I.  Consider the quadratic equation in standard form…


    Let’s examine graphs of a quadratic equation in which “a” and “c” are held constant and “b” is varied.


    In the figure below, the beginning equation is…



and “b” is varied.



From the graphs, you can draw several conclusions….


1. When “b” is -3 and 3, the parabolas intercept the x-axis in two places each. In other words, each parabola has two zero’s or

    two real solutions. These can be found by setting the equation equal to 0 and solving for x. For example….

(Solved using the Quadratic Formula.)


2. When “b” 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 example….


3. When “b” 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 below….


4. When “b” 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 “b” is 0 the equation becomes…


When viewed in vertex form, the equation becomes…


From the above equation, the vertex can easily be identified as (0, 1).



II.  Next, let’s find the vertices of each of the graphs using the vertex formula for quadratic equations in standard form….


     The work will be given for the first problem only….


     * If  b = 3….



        Therefore, the vertex is (-1.5, -1.25).





(-1.5, -1.25)

(1.5, -1.25)

(-1, 0)

(1, 0)

(-0.5, 0.75)

(0.5, 0.75)

(0, 1)



III. Next, consider…


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 for…



















       Also, notice that the purple parabola is a reflection of the graph whose “b” is 0.


       Thus, it can be stated that the reflection of a graph of a quadratic equation in which “b” is 0 will contain the vertices of

       the parabolas created when “b” is varied and “a” and “c” are held constant in the original equation.




                 If the hypotheses is true, then…


should be the reflection of…



and should contain the vertices of the parabolas created when only “b” is varied in the original equation.


Red: b = -3, 3

Blue: b = -2, 2

Green: b = -1, 1