Quadratic functions have traditionally occupied a large part of secondary mathematics. Mostly, the emphasis has been on finding the zeros, or roots. That is, given the quadrtic function

,

solve for the values of *x* that satisfy the quadratic equation

.

If we let *a* = *c* = 1, we get the equation

,

which is graphed in the *x* - *b* plane below:

If we set *b* equal to a particular value, such as 3, and look at the two graphs together, we see that the graph of *b* = 3 intersects the graph of *x*^{2} + *bx* +1 = 0 at two points. These points correspond to the roots of the quadratic equation *x*^{2} + 3*x* +1 = 0, as shown:

Click here to see an animation of -7 < *b* < 7. This will show the values of *b* for which there are 0, 1 or 2 real roots of *x*^{2} + *bx* + 1 = 0.

Notice that for *b* = 2 there is exactly one negative real root, and for *b* = -2 there is exactly one positive real root. This is clear, because solving the quadratic equations

*x*^{2} + 2*x* + 1 = 0 and *x*^{2} - 2*x* + 1 = 0

is equivalent to solving the equations

(*x + *1)^{2} = 0 and (*x - *1)^{2} = 0,

respectively, yielding one (repeated) real root in each case. Otherwise, for -2 < *b* < 2, there are no real roots for *x*^{2} + *bx* +1 = 0, and for *b* > 2 or *b* < -2 there are 2 real roots.

In the case above, we fixed the value of *c* at 1. The following illustration shows a family of hyperbolas formed in the *x - b* plane by letting *c* = 1, 3, 5, 7, 0, -1, -3, -5, and -7.

For *c* = 0 , we get the asymptotes for all of the hyperbolas. Otherwise, for *c* > 0 , varying the value of *b* as before, we have 0, 1 or 2 real roots as given by the intersection of the graph of *b* = *n* and the graph of the particular (vertically oriented) hyperbola for the given value of c.

For c < 0 the function will have 2 real roots, as given by the intersection of the graph of *b* = *n* and the graph of the particular (horizontally oriented) hyperbola for the given value of c.