Precalc 9.1 Parabolas
For each parabola, find an equation for the axis of
symmetry and the coordinates of the vertex. State
whether the parabola opens up or down, and whether
the y-coordinate of the vertex is the minimum or
maximum value of the function.
A parabola is defined in terms of a fixed point,
called the focus, and a fixed line, called the directrix.
A parabola is the set of all points P(x,y)
in the plane whose distance to the focus
equals its distance to the directrix.
directrix
Mar 30­7:52 AM
focus
axis of symmetry
Mar 30­7:55 AM
Vertical Directrix
Horizontal Directrix
Standard Equation of a parabola with its vertex at
the origin is
p > 0: opens upward
p < 0: opens downward
focus: (0, p)
directrix: y = –p
axis of symmetry: y-axis
Standard Equation of a parabola with its vertex at
the origin is
p > 0: opens right
p < 0: opens left
focus: (p, 0)
directrix: x = –p
axis of symmetry: x-axis
Mar 30­8:03 AM
The standard form of the equation of a parabola with vertex at (h,k) is as follows:
Mar 30­8:07 AM
Graph
directrix.
. Label the vertex, focus, and
vertical axis directrix: y = k ­ p (x ­ h)2 = 4p(y ­ k), p ≠ 0
horizontal axis; directrix: x = h ­ p (y ­ k)2 = 4p(x ­ h), p ≠ 0
The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0,0), the equation takes one of these forms:
Vertical Axis:
Horizontal Axis:
x2 = 4py
y2 = 4px
Mar 9­7:42 AM
Mar 30­8:09 AM
1
Identify the focus & the directrix
Write the standard equation of the parabola
with its vertex at the origin and the directrix
y = -6.
Graph y = 1/8x2
Focus (­3,0)
Directrix y = 2
Mar 9­8:27 AM
Standard Equation of a Translated Parabola
Horizontal Directrix:
vertex: (h, k)
focus: (h, k + p)
directrix: y = k – p
axis of symmetry: x = h
Mar 30­8:12 AM
Write the standard equation of the parabola with a
focus at F(-3,2) and directrix y = 4.
Mar 30­8:10 AM
Standard Equation of a Translated Parabola
Vertical Directrix:
vertex: (h, k)
focus: (h + p, k)
directrix: x = h - p
axis of symmetry: y = k
Mar 30­8:14 AM
Write the standard equation of the parabola
with its focus at F(-6,4) and directrix x = 2.
I graphed parabola from given. Find vertex & p, then write equation.
Mar 30­8:16 AM
Mar 30­8:20 AM
2
Graph the parabola
vertex, focus, and directrix.
-4(
. Label the
) ( )-4
Isolate the y-terms
Complete the square
vertex: (h, k) = (1,-2)
Find p:
so, p = 1
focus: = (2,-2)
directrix: x = 0
Mar 30­8:22 AM
Mar 30­8:29 AM
Graph the parabola x2 – 6x + 6y + 18 = 0.
Label the vertex, focus, and directrix.
Mar 30­8:33 AM
Mar 9­7:54 AM
3