Frames:

2. Quadratic equations

2.1 The general form of quadratic equations

A conic in the two-dimensional plane is the solution set of an equation of the form

ax2 + bxy + cy2 + dx + ey + f = 0                           

for real parameters a, b, c, d, e and f, where at least one of a, b and c is not zero.

René Descartes (1596 - 1650) was the first to make a classification of curves and their algebraic formulae. We will not make an analysis here, but it is known that the value of the discriminant b2 – 4ac (the same one we know so well, but with different meanings!), determines the type of conic:

  • If b2 – 4ac > 0, then the graph is a hyperbola (or two intersecting lines).
  • If b2 – 4ac = 0, then the graph is a parabola (or two parallel lines, one line or no curve).
  • If b2 – 4ac < 0, then the graph is an ellipse (if b = 0 and a = c, the graph is a circle).

Let's investigate it! Open the Quadratic Equations applet:  

  1. In the applet, start with an open investigation: play with the values of the parameters and try to make at least one example of each of the different types of conics. For what values of the parameters are these different quadratic graphs formed?

  2. Investigate the following:
    Let d = e = f = 1. Vary the values of a, b, and c to consider the following cases:
    b2 – 4ac < 0, b2 – 4ac = 0, and b2 – 4ac > 0. Identify which conics fit into each category.
    Once you have a graph, also change d, e and f and convince yourself that these parameters do not make new shapes, but only transform an existing shape.

  3. Convince yourself that the functions we study at school:
    1. The straight line is the special case dx + ey + f = 0 (i.e. a = b = c = 0).
    2. The parabola is the special case ax2 + dx + ey + f = 0 (i.e. b = c = 0).
    3. The circle is the special case ax2 + cy2 + f = 0 with a = c, or ax2 + cy2 +dx + ey + f = 0.
    4. The hyperbola is the very special case bxy + f = 0 (i.e. a = c = d = e = 0).
  4. Check the following and explain why:
    1. a = 1, b = 2, c = 1 (and all the other parameters 0) is a line. Then, if also f = 1, it is nothing and if f = -1 it is two parallel lines.
    2. a = 1, b = 2, c = 1 is a line. Then, if also d = 1 it is a parabola, and if d = 1 and e = 1 it is two parallel lines, and if d = 1 and e = 2 it is a parabola.

  5. Start with just a = 1 and all the other parameters 0. It is the Y-axis. Now add d = 1, it is two vertical lines.  Now change just c and explain why c = 1 is a circle, and c = - 1 is a hyperbola.

  6. Start with just a = 1. It is the Y-axis.
    Now change c: c = 1 gives nothing, c = -1 gives two intersecting lines.
    Now add just f = -1 – why is it a hyperbola?