We have to get it in the standard form:
where (h,k) is the vertex, and |p| is the distance
from the vertex to the focus and also the distance
from the vertex to the directrix. If p is positive
the parabola opens upward and if p is negative the
parabola opens downward.
The left side of
is already in that form. We factor out -24
Reduce the fraction:
Now we can compare it to
and see that
-h=+6 so h=-6
-k = -17/8 so k = 17/8
4p = -24, so p = -6
The vertex is (h,k) or (-6,17/8).
We sketch the parabola. It goes through the
vertex (-6,17/8) as opens downward since p
is a negative number.
The focus (which is inside the parabola is |p| or 6
units below the vertex. It has the same x-coordinate -6
So to find its y-coordinate we subtract 6 from the
y-coordinate of the vertex 17/8:
So the focus is (-6,-31/8)
Finally we will draw the directrix, which is a horizontal
line (in green below) which is |p| = 6 units outside (above)
the vertex. We determine how far above the x-axis that is
by adding 6 to the y-coordinate of the vertex 17/8:
So the directrix is a horizontal line y = 65/8
The axis of symmetry is the vertical line (in blue below)
through both the vertex and the focus which cuts the
parabola in two. It is the line x=(the x-coordinate of
the vertex and focus). In this case the equation of the
axis of symmetry is x=-6:
Edwin
