Exploring Quadratics From a Different Perspective: Using Alternative Planes

By: Teo Paoletti

The x-c Plane

We initially look at the equation .  The graph is

We can see that the graph is simply an upside down parabola.  This makes sense as we can transform our function to be .  Hence the zeros of our new function are x = 0 and x = -1.  Since the equation in the x-c plane has only an x-squared and x term, our graph will only stretch or shrink according to the value of b.  The zeros will always correspond to x = 0 and x = -b.  We can see this in the animation below.

The x-a Plane

We first look at the x-a plane through the graph of .  Unlike the x-c plane which also gave us a parabola, this plane provides a dramatically different picture.

The graph appears to approach 0 and x goes to infinity and negative infinity.  The graph seems to have a maximum around -1, and a vertical asymptote at x = 0 (although we would need a zoomed out picture to see that).  The graph also appears to only have one zero at x = - ½ .  The zero is easy to explain, because if a = 0, the equation simply becomes 0 = 2x + 1 which has a solution of – ½ .  The horizontal asymptote can be explained by the fact that if x = 0, weĠd have the equation 0 = 1, which is a contradiction.  Hence the graph does not exist at x = 0.  It is worth noting that if we were to eliminate c, and graph , then when x = 0, weĠd have 0=0 which is true.  However, there is still a vertical asymptote at x = 0, but now the two sides go in opposite directions.

This graph makes sense though because if we were to solve  we get .

Now going back to our original graph, if we insert the line a = -1, we see that the line crosses the graph at exactly two points.  We realize that these two points would correspond to the solutions of the equation .

Similarly if we slide the line y = n up and down we see that the line crosses the graph at either 0, 1, or 2 places.  If n > 1 the lines never intersect, meaning there are no solutions to our original equation.  If n = 1 or 0 then there is only one solution to our original equation.  And if 0 < n < 1 or n < 0, then there are two solutions to the original equation.  Hence we can use the x-a plane to explore the zeros of our initial equation as well.