Quadratics - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {25}
Exercise 13
Put the parabola y 2 − 2y = 4x − 5 into the above standard form (for a directrix parallel to
the y-axis) and thus find its vertex, focus and directrix, and sketch its graph.
Parametric equations of a parabola
Since every parabola is congruent to x 2 = 4ay, for some choice of a > 0, we can study the
geometry of every parabola by confining ourselves to this one! This shows the power of
both coordinate geometry and transformations.
Imagine a particle moving in the plane along a curve C as shown.
y
P(x,y) = (f(t),g(t))
0
x
The x- and y-coordinates of the particle can be thought of as functions of a new variable
t, and so we can write x = f (t), y = g (t), where f , g are functions of t. In some
physical problems t is thought of as time.
The equations
x = f (t)
y = g (t)
are called the parametric equations of the point P(x, y) and the variable t is called a
parameter.
It is often very useful to take a cartesian equation y = F (x) and introduce a parameter t
so that the x- and y-coordinates of any point on the curve can be expressed in terms of
this parameter.