Section 9.1

Conics

OBJECTIVE 1

.

x

y

Focus (0,p)

Vertex (h,k)

Geometric Definition of a Parabola: The collection of all the points P(x,y) in a a plane that are the same distance from a fixed point, the focus, as they are from a fixed line called the directrix.

P

x

y

Focus (0,p)

Directrix

Vertex (h,k)

p

p

2p

And the equation is…

As you can plainly see the distance from the

focus to the vertex is a and is the same distance

from the vertex to the directrix! Neato!

y = -p

x 2 4 py or y 1

4 px 2

x

y

Focus (0,-p)

Directrix y a

Vertex (h,k)

p

p

And the equation is…

x 2 4 py or y 1

4 px 2

x

y

Focus (p,0)

Vertex (h,k)

x aDirectrix p p

2p

And the equation is…

y 2 4 px or x 1

4 py 2

x

y

Focus (-p,0)Vertex

(h,k)

x aDirectrix p p

And the equation is…

y 2 4 px or x 1

4 py 2

STANDARD FORMS

Vertex at ( , )

Opens up

h k

Vertex at ( , )

Opens down

h k

Vertex at ( , )

Opens right

h k

Vertex at ( , )

Opens left

h k

I like to call standard form “Good Graphing Form”

1) (x h)2 4 p(y k) or y 1

4 p(x h)2 k

2) (x h)2 4 p(y k) or y 1

4 p(x h)2 k

3) (y k)2 4 p(x h) or x 1

4 p(y k)2 h

4) (y k)2 4 p(x h) or x 1

4 p(y k)2 h

A Couple More Things…………

A parabola that opens up/down:

1) Has a vertex at (h , k)

2) Has an Axis of Symmetry at x = h

3) Coordinates of focus (h , k+p)

4) Equation of Directrix y = k-p

A parabola that opens right/left:

1) Has a vertex at (h , k)

2) Has an Axis of Symmetry at y = k

3) Coordinates of focus (h+p , k)

4) Equation of Directrix x = h-p

General Form of any Parabola

2 2 0Ax By Cx Dy E

*Where either A or B is zero!

* You will use the “Completing the Square” method to go from the General Form to Standard Form,

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.

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.

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 + 72y2 + 12x - 10y - 47 = 0 General form

(y - 5)2 = 4(-3)(x - 6)(y - 5)2 = -12(x - 6)

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 equationand 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 - 48x2 + 4x + 8y + 52 = 0 General form

3.6.10

Finding the Equations of Parabolas

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

Standardform

3.6.11

Analyzing a Parabola

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

From the graph, the vertex is at the origin, (0,0), and the directrix is 2 units away from the vertex.

 The parabola opens up, so the equation is in form. Since p = 2 , the equation is

Example #2 Writing the equation of a parabola

x py2 4

p 2

yx )2(42

yx 82 (Larson, Boswell, Kanold & Stiff, 2005)

x 2 4 py

#10 Write the standard form of the equation of the parabola with the given focus or directrix with the vertex at (0,0). Focus

)3,0(

Since the focus has to be inside the parabola and lie on the axis of symmetry, this parabola opens up, and is the form

x 2 4 pyThe distance p is the distance from the vertex to the focus, or in this case 3.

x y x y2 24 3 12 ( )

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) = 0x + 3 = 0 or x + 2 = 0 x = -3 x = -2The result is two vertical,parallel lines. 3.6.13