1. The Parabola 3.6 Chapter 3 Conics 3.6.1
MATHPOWERTM 12, WESTERN EDITION
3.6.1

2. The Parabola
The parabola is the locus of all points in a plane that are
the same distance from a line in the plane, the directrix,
as from a fixed point in the plane, the focus.
Point Focus = Point Directrix
PF = PD
The parabola has one axis of
symmetry, which intersects
the parabola at its vertex.
| p |
The distance from the
vertex to the focus is | p |.
The distance from the
directrix to the vertex is also | p |.
3.6.2

3. The Standard Form of the Equation of a Parabola with Vertex (0, 0)
vertex (0, 0) and focus on the x-axis
is y2 = 4px.
The coordinates of the focus are (p, 0).
The equation of the directrix is x = -p.
If p > 0, the parabola opens right.
If p < 0, the parabola opens left.
3.6.3

4. The Standard Form of the Equation of a Parabola with Vertex (0, 0)
vertex (0, 0) and focus on the y-axis
is x2 = 4py.
The coordinates of the focus are (0, p).
The equation of the directrix is y = -p.
If p > 0, the parabola opens up.
If p < 0, the parabola opens down.
3.6.4

5. A parabola has the equation y2 = -8x. Sketch the
Sketching a Parabola
A parabola has the equation y2 = -8x. Sketch the
parabola showing the coordinates of the focus and
the equation of the directrix.
The vertex of the parabola is (0, 0).
The focus is on the x-axis.
Therefore, the standard equation is y2 = 4px.
Hence, 4p = -8
p = -2.
The coordinates of the
focus are (-2, 0).
F(-2, 0)
The equation of the
directrix is x = -p,
therefore, x = 2.
x = 2
3.6.5

6. Finding the Equation of a Parabola with Vertex (0, 0)
A parabola has vertex (0, 0) and the focus on an axis.
Write the equation of each parabola.
a) The focus is (-6, 0).
Since the focus is (-6, 0), the equation of the parabola is y2 = 4px.
p is equal to the distance from the vertex to the focus, therefore p = -6.
The equation of the parabola is y2 = -24x.
b) The directrix is defined by x = 5.
Since the focus is on the x-axis, the equation of the parabola is y2 = 4px.
The equation of the directrix is x = -p, therefore -p = 5 or p = -5.
The equation of the parabola is y2 = -20x.
c) The focus is (0, 3).
Since the focus is (0, 3), the equation of the parabola is x2 = 4py.
p is equal to the distance from the vertex to the focus, therefore p = 3.
The equation of the parabola is x2 = 12y.
3.6.6

7. The Standard Form of the Equation with Vertex (h, k)
For a parabola with the axis of symmetry parallel to
the y-axis and vertex at (h, k):
The equation of the axis of symmetry is x = h.
The coordinates of the focus are (h, k + p).
The equation of the directrix is y = k - p.
When p is positive, the parabola opens upward.
When p is negative, the parabola opens downward.
The standard form for parabolas
parallel to the y-axis is:
(x - h)2 = 4p(y - k)
The general form of the parabola
is Ax2 + Cy2 + Dx + Ey + F = 0
where A = 0 or C = 0.
3.6.7

8. The Standard Form of the Equation with Vertex (h, k)
For a parabola with an axis of symmetry
parallel to the x-axis and a vertex at (h, k):
The equation of the axis of symmetry is y = k.
The coordinates of the focus are (h + p, k).
The equation of the directrix is x = h - p.
When p is positive, the parabola
opens to the right.
When p is negative, the parabola
opens to the left.
The standard form for parabolas
parallel to the x-axis is:
(y - k)2 = 4p(x - h)
3.6.8

9. Finding the Equations of Parabolas
Write the equation of the parabola with a focus at (3, 5) and
the directrix at x = 9, in standard form and general form
The distance from the focus to the directrix is 6 units,
therefore, 2p = -6, p = -3. Thus, the vertex is (6, 5).
The axis of symmetry is parallel to the x-axis:
(y - k)2 = 4p(x - h)
h = 6 and k = 5
(y - 5)2 = 4(-3)(x - 6)
(y - 5)2 = -12(x - 6)
(6, 5)
Standard form
y2 - 10y + 25 = -12x + 72
y2 + 12x - 10y - 47 = 0
General form
3.6.9

10. Finding the Equations of Parabolas
Find the equation of the parabola that has a minimum at
(-2, 6) and passes through the point (2, 8).
The axis of symmetry is parallel to the y-axis.
The vertex is (-2, 6), therefore, h = -2 and k = 6.
Substitute into the standard form of the equation
and solve for p:
(x - h)2 = 4p(y - k)
x = 2 and y = 8
(2 - (-2))2 = 4p(8 - 6)
16 = 8p
2 = p
(x - h)2 = 4p(y - k)
(x - (-2))2 = 4(2)(y - 6)
(x + 2)2 = 8(y - 6)
Standard form
x2 + 4x + 4 = 8y - 48
x2 + 4x + 8y + 52 = 0
General form
3.6.10

11. Find the coordinates of the vertex and focus,
Analyzing a Parabola
Find the coordinates of the vertex and focus,
the equation of the directrix, the axis of symmetry,
and the direction of opening of y2 - 8x - 2y - 15 = 0.
4p = 8
p = 2
y2 - 8x - 2y - 15 = 0
y2 - 2y + _____ = 8x _____ 1 1
(y - 1)2 = 8x + 16
(y - 1)2 = 8(x + 2)
Standard
form
The vertex is (-2, 1).
The focus is (0, 1).
The equation of the directrix is x + 4 = 0.
The axis of symmetry is y - 1 = 0.
The parabola opens to the right.
3.6.11

12. Graphing a Parabola y2 - 10x + 6y - 11 = 0
y2 + 6y + _____ = 10x _____ 9 9
(y + 3)2 = 10x + 20
(y + 3)2 = 10(x + 2)
3.6.12

13. General Effects of the Parameters A and C
When A x C = 0, the resulting
conic is an parabola.
When A is zero:
If C is positive,
the parabola opens to the left.
If C is negative,
the parabola opens to the right.
When C is zero:
If A is positive,
the parabola opens up.
If A is negative,
the parabola opens down.
When A = D = 0, or when C = E = 0,
a degenerate occurs.
E.g., x2 + 5x + 6 = 0
x2 + 5x + 6 = 0
(x + 3)(x + 2) = 0
x + 3 = 0 or x + 2 = 0
x = x = -2
The result is two vertical,
parallel lines.
3.6.13

14. Assignment
3.6.14