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Quadratic functions

by Zack Kroll

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We want to investigate the graph of .

First we graph this equation where c =1. The graph for the equation:

 

These graphs are interesting because one of the two branches of the hyperbola can be rotated 180 degrees to lay on top of the other.

If we change the value of c to odd values up to 9 and graph them this is what we get. As the constant of the equation changes, the branches move further away from one another. The bases (loops) of the graphs are also wider as the value of c increases. Because we are only changing the constant and not any of the coefficients or exponents in the equations the asymptotes at y=0 and y=-x exist for each of the graphs.

 

When we substitute different values in for c (e.g. 3, 5, 7, and 9) we create graphs that look like this.

If we take any particular value of b, say b = 5, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.

 

For each value of b we select, we get a horizontal line. It is clear on a single graph that we get two negative real roots of the original equation when b > 2, one negative real root when b = 2, no real roots for -2 < b < 2, One positive real root when b = -2, and two positive real roots when b < -2. As we can see there are no real roots between b = 2 and b = -2 as the graphs do not appear on the coordinate plane between those two values.

By changing the value of c from 1 to -1, we are able to change the orientation of the entire graph. Instead of the graph opening up and down (purple) it now opens left and right (blue).

If we continue this for each of the other equations that we have previously graphed then we get this (e.g. -3, -5, -7, and -9). The graphs with positive c values are in blue while those with negative c values are in green.

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