College Algebra & Trigonometry, 2018a

5.2. THE EQUATION OF THE PARABOLA 275
Here’s another example:
Express the equation in standard form and determine the vertex, focus and directrix
of the following parabola.
y 2 − 2y − 1+8x =0
We know that this parabola will be either right or left facing.
First, let’s move all the terms not containing y to the right-hand side but leave
space to complete the square.
y 2 − 2y = −8x +1
Add 1 to both sides to complete the square:
y 2 − 2y +1=−8x +1+1
(y − 1) 2 = −8x +2
Factor out the coefficient of the x variable:
(y − 1) 2 = −8(x − 0.25)
So, this is a left-facing parabola with a vertex at the point (0.25, 1). To find the
focal distance, we say that 4p = −8, sop = −2.
Since it is left-facing, the focus will be a distance of 2 units to the left of the vertex
at the point (−1.75, 1) and the directrix will be a distance of 2 units to the right of
the vertex at the line x =2.25.