Christa Marie Nathe

In this investigation we will look at the basic quadric equation ax2+bx+c=y and the patterns of roots that arise from graphing said equation.  The values for a, b and c all play a vital role in the graphical representation of the equation. Below one will find several graphs of the equation y=x2+bx+1, where the values of b are changed.  Looking at the graph, which values of b correspond to which parabola?

When b is negative 3(red), 2(blue) and 1(green) the parabola lies in the first and fourth quadrant. Alternatively, when b is positive 3(pink), 2(gray) and 1(yellow) the parabola occupies the second and fourth quadrants. What happens when b is zero? That is the graph, which is represented by the light blue parabola.  It does not cross the x-axis into another quadrant.  All of these graphs have one characteristic in common; they all cross the y-axis at the same point. The value of c=1 does not change, and determines the point where the parabola will cross the y-axis.

Now that we have identified which b value represents which parabola, lets look at where those graphs cross the x-axis and its significance.  Where the graph crosses the x-axis is where there is a root of the equation. For which value of b does the graph cross the x-axis more than once? When b is -3 or 3. Therefore there exists two real roots positive and negative respectively.  What happens when b is -2 and 2? The parabola is tangent to the x-axis and hence has only one real root.  When are there no real roots for the equation graphed? Yes, when b is -1 or 1.

Looking at the vertices of the parabolas above, what significance do they relay? If the vertices are traced and connected they will form an upside down parabola where y=-x2+1 as depicted below.