10.2a Introduction to Conics - Parabola
Conics – conic section is the intersection of a plane and a right circular cone, see the figures below.
II. Parabola
Definition: A parabola is the set of all points (x, y) in the plane that are equidistant from a fixed line (called
directrix), and a fixed point (focus) not on the line.
The vertex is the midpoint between the focus and directrix.
The axis (axis of symmetry) is the line through the focus and perpendicular to the directrix.
Axis of symmetry
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Standard form of the equation of a parabola with vertices at (h, k):
2
1. Vertical axis: ( x  h )  4 p ( y  k ) where p  0
a. axis: x = h
b. vertex: (h, k)
c. focus : (h, k+p)
d. directrix: y = k – p
2
2. Horizontal axis: ( y  k )  4 p ( x  h ) where p  0
a. axis: y = k
b. vertex: (h, k)
c. focus : (h+p, k)
d. directrix: x = k – p
Example 1: Find the vertex, focus, and directrix of the parabola given by x2 2 y
Solution:
First, write the equation in the standard form of the parabola, this is a parabola with vertical axis:
( x 0)2 4 p ( y 0)
Then compare with the vertical axis equation in 1, we have h=0, k= 0, 4 p2
Thus
a. axis: x = 0
b. vertex: (0, 0)
c. focus: (0, 0+1/2) = (0, 1/2)
d. directrix: y = 0-1/2 = -1/2
Focus: (0, ½)
axis: x = 0
 p 12
Vertex (0, 0)
Directrix x = -1/2
Example 2: Given the equation: ( y 2)2 8( x 3) , find the axis, vertex, focus and directrix; then sketch the
parabola.
Solution: First compare with the standard form of the equation of parabola, this parabola has horizontal line
as axis, which is y  2, 4p  8  p  2
Directrix x = 4
Thus
a.
b.
c.
d.
Focus: (0, 2)
axis: y = 2
vertex: (3, 2)
focus: (2+(–2), 3) = (0, 2)
directrix: x = 2 – (– 2) = 4
axis: y =2
vertex: (3, 2)
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Remember this:
For vertical axis and if p > 0, then the parabola opens up to the positive y-axis; if p < 0, then the parabola
opens up to the negative y-axis.
For horizontal axis and if p > 0, then the parabola opens up to the positive x-axis; if p < 0, then the parabola
opens up to the negative x-axis.
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