Let's consider the simplest parabola, which can be represented by the space of points in over an arbitrary field with that satisfies the equation . Note that a general affine transformation on this parabola will result in a general parabola.
To compute the tangent line of the parabola at the point we pick another point on the parabola, so that the line joining and has vector equation
for . Noting that , we use the fact that to obtain
which simplifies to
This line is called the secant line or the secant chord. If , then and the secant chord has equation ; this line is then called the tangent line of the parabola at the point . We now have a methodology for exactly computing tangent lines to parabolas which avoids limiting processes. Note that this also implies that the vertex of the parabola, i.e. the point at which the tangent line of the parabola takes the form for a constant , is
As an example, we consider the general affine transformation given by
The vertex is mapped by to , which is the vertex of the translated parabola. I will leave it as an exercise to figure out