The document covers the properties and equations of parabolas, detailing their definition as the locus of points equidistant from a focus and a directrix. It presents standard forms of parabolic equations for different orientations, methods for deriving them from properties such as focus and vertex, and the implications of parameters on the parabola's orientation. Additionally, the document includes examples and exercises to reinforce the concepts discussed.

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

2. 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
The Parabola

3. The Standard Form of the Equation of a Parabola with Vertex (0, 0)
The equation of a parabola with
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 equation of a parabola with
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
The Standard Form of the Equation of a Parabola with Vertex (0, 0)

5. 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).
The equation of the
directrix is x = -p,
therefore, x = 2.
F(-2, 0)
x = 2
Sketching a Parabola
3.6.5

6. A parabola has vertex (0, 0) and the focus on an axis.
Write the equation of each parabola.
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.
The equation of the directrix is x = -p, therefore -p = 5 or p = -5.
The equation of the parabola is y2
= -20x.
3.6.6
Finding the Equation of a Parabola with Vertex (0, 0)
Since the focus is on the x-axis, the equation of the parabola is y2
= 4px.
c) The focus is (0, 3).
a) The focus is (-6, 0).
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.

7. 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
The Standard Form of the Equation with Vertex (h, k)

8. 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.
• The standard form for parabolas
parallel to the x-axis is:
(y - k)2
= 4p(x - h)
• When p is negative, the parabola
opens to the left.
• When p is positive, the parabola
opens to the right.
3.6.8
The Standard Form of the Equation with Vertex (h, k)

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).
(6, 5)
The axis of symmetry is parallel to the x-axis:
(y - k)2
= 4p(x - h) h = 6 and k = 5
Standard form
y2
- 10y + 25 = -12x + 72
y2
+ 12x - 10y - 47 = 0 General form
(y - 5)2
= 4(-3)(x - 6)
(y - 5)2
= -12(x - 6)
3.6.9

10. 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)
(2 - (-2))2
= 4p(8 - 6)
16 = 8p
2 = p
x = 2 and y = 8
(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
Finding the Equations of Parabolas

11. 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.
y2
- 8x - 2y - 15 = 0
y2
- 2y + _____ = 8x + 15 + _____1 1
(y - 1)2
= 8x + 16
(y - 1)2
= 8(x + 2)
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.
4p = 8
p = 2
Standard
form
3.6.11
Analyzing a Parabola

12. Graphing a Parabola
y2
- 10x + 6y - 11 = 0
9 9y2
+ 6y + _____ = 10x + 11 + _____
(y + 3)2
= 10x + 20
(y + 3)2
= 10(x + 2)
y + 3 = ± 10(x + 2)
y = ± 10(x + 2) − 3
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 A = D = 0, or when C = E = 0,
a degenerate occurs.
When C is zero:
If A is positive,
the parabola opens up.
If A is negative,
the parabola opens down.
E.g., x2
+ 5x + 6 = 0 x2
+ 5x + 6 = 0
(x + 3)(x + 2) = 0
x + 3 = 0 or x + 2 = 0
x = -3 x = -2
The result is two vertical,
parallel lines. 3.6.13

14. Suggested Questions:
Pages 167-169
A 1, 3, 5, 7, 8 (general form)
3.6.14
B 9, 12, 13, 14, 16, 18,
27, 28, 29, 46, 47