Characterizing parabolas and their graphs

Summary

Characterizing a parabola means gathering, from its equation, the data that suffice to draw it: the direction in which it opens, the vertex, the axis of symmetry, the focus, the directrix and the intercepts with the coordinate axes. This lesson shows how those data are obtained from the vertex form and from the polynomial form, derives the general formulas for the vertex, the focus and the directrix in terms of the coefficients, relates the intercepts with the $x$ -axis to the discriminant and takes advantage of the symmetry of the curve to shorten the computations. It ends with an orderly procedure for sketching the graph.

Learning objectives

Obtain the vertex, the focus, the directrix and the axis of a parabola given in vertex form or in polynomial form.
Derive, by completing the square, the general formulas for the elements of the parabola $y = ax^2 + bx + c$ .
Determine the intercepts with the axes and decide, by means of the discriminant, whether the parabola intersects the $x$ -axis.
Use the symmetry with respect to the axis to find the vertex and sketch the graph with the information obtained.

The two forms of the equation

In the lesson on the equation of parabolas it was shown that the parabola with a vertical axis, vertex at $(h, k)$ , focus $(h, k + f)$ and directrix $y = k - f$ , where $f \neq 0$ , satisfies the equation that we shall call here the vertex form; the absolute value of $f$ is the focal distance, and its sign indicates which way the curve opens:

(x - h)^2 = 4f(y - k)

It was also proved, in the theorem on the graphs of quadratic polynomials, that solving for $y$ turns this equation into a quadratic polynomial and that, conversely, the graph of every quadratic polynomial is a parabola with a vertical axis. The equation below will be called the polynomial form:

y = ax^2 + bx + c, \quad a \neq 0

Both describe the same curve, but they do not display the same information: the vertex form displays the geometry (vertex, focus, directrix), whereas the polynomial form displays the intercept with the $y$ -axis, which is $(0, c)$ , and allows the intercepts with the $x$ -axis to be computed by solving a quadratic equation. Characterizing a parabola consists in gathering the following data, with which its graph is sketched:

The direction of opening: upwards or downwards.
The vertex and the axis of symmetry.
The focus and the directrix.
The intercepts with the $x$ -axis, if any, and the intercept with the $y$ -axis.