College Algebra & Trigonometry, 2018a

266 CHAPTER 5. CONIC SECTIONS - CIRCLE AND PARABOLA
5.2 The Equation of the Parabola
The equation of the parabola is often given in a number of different forms. One
of the simplest of these forms is:
(x − h) 2 =4p(y − k)
A parabola is defined as the locus (or collection) of points equidistant from a
given point (the focus) and a given line (the directrix). Another important point
is the vertex or turning point of the parabola. If the equation of a parabola is
given in standard form then the vertex will be (h, k). The focus will be a distance
of p units from the vertex within the curve of the parabola and the directrix will
be a distance of p units from the vertex outside the curve of the parabola. This
value (p) is called the focal distance.
Focus: (h, k + p)
Vertex: (h, k)
Directrix: y = k − p
Any point on the curve of the parabola is equidistant from the focus (h, k + p)
and the directrix (h, k − p). Notice that the focus is a point and is identified with
the coordinates of the point while the directrix is a line and is identified with the
equation for that line.