A quadratic function is a function of the form *y* = *ax*^{2} + *bx* + *c*, where *a*≠ 0, and *a*, *b*, and *c* are real numbers.

### Intercepts of a Quadratic Function

The *y*-intercept is given by *x* = 0: *y* = *a*(0^{2}) + *b*(0) + *c* = *c*. Thus, the *y*-intercept is (0, *c*).

The *x*-intercept is given by *y* = 0: 0 = *ax*^{2} + *bx* + *c*. Thus, the *x*-intercept(s) can be found by factoring or by using the quadratic formula.

In addition, the discriminant gives the number of *x*-intercepts of a quadratic function, because it gives us the number of solutions to *ax*^{2} + *bx* + *c* = 0. If *b*^{2} -4*ac* > 0, there are 2 solutions to *ax*^{2} + *bx* + *c* = 0 and consequently 2 *x*-intercepts. If *b*^{2} - 4*ac* = 0, there is 1 solution to *ax*^{2} + *bx* + *c* = 0, and consequently 1 *x*-intercept. If *b*^{2} -4*ac* < 0, there are no solutions to *ax*^{2} + *bx* + *c* = 0, and consequently no *x*-intercepts. The graph of the function does not cross the *x*-axis; either the vertex of the parabola is above the *x*-axis and the parabola opens upward, or the vertex is below the *x*-axis and the parabola opens downward.

### Completing the Square

A quadratic function in the form *y* = *ax*^{2} + *bx* + *c* is not always simple to graph. We do not know the vertex or the axis of symmetry simply by looking at the equation. To make the function easier to graph, we need to convert it to the form *y* = *a*(*x* - *h*)^{2} + *k*. We do this by completing the square: adding and subtracting a constant to create a *perfect square trinomial* within our equation.

A perfect square trinomial is of the form *x*^{2} +2*dx* + *d*^{2}. In order to "create" a perfect square trinomial within our equation, we must find *d*. To find *d*, divide *b* by 2*a*. Then square *d* and multiply by *a*, and add and subtract *ad*^{2} to the equation (we must add and subtract in order to maintain the original equation). We now have an equation of the form *y* = *ax*^{2} +2*adx* + *ad*^{2} - *ad*^{2} + *c*. Factor *ax*^{2} +2*adx* + *ad*^{2} into *a*(*x* + *d* )^{2}, and simplify - *ad*^{2} + *c*.