Assignment Three

Quadratic Equations

Charles Meyer


For this investigation, I will compare different graphs of y = x2 + bx + 1.  I am concerned about how the graph changes as values of b change.  The values of b that I really want to concentrate on are b = -3, -2, -1, 0, 1, 2, 3


The question is then posed as to what does the changes in b do to the graph.  It is noted that no matter what, the graphs pass through y = 1 at all values of b.


The graph also helps us to see the roots of each equation. The graph's intersection with the x-axis, tells the number of roots and the value of those roots of the equation.  If b < -2 or b > 2 then the equation has two roots.  Those equations when b  is less of -2 then both roots are negative, equations where b is greater than 2, then both roots are positive.  Equations where b is equal to -2 or 2, then the equation has only one root.  Finally b is greater than -2 but less than 2 then there are no real roots to the equation.


Another interesting part of this graph is the vertices of each equation.  If these vertices are connected, a new parabola, facing downward is formed.

In this investigation, it was discovered that given an equation of y = ax2 + bx + c, if a and c remain constant but b varies then the locus of the vertices of all parabolas will also form a parabola.  This investigation would be very beneficial to a 8th or 9th grade Algebra student in learning the value of graphing equations.


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