10.110.110.110.1

Conic Sections and ParabolasConic Sections and Parabolas

Quick Review 2

2

1. Find the distance between ( 1,2) and (3, 4).

2. Solve for in terms of . 2 6

3. Complete the square to rewrite the equation in vertex form.

2 5

4. Find the vertex and axis of the graph of

y x y x

y x x

f

2

2

( ) 2( 1) 3.

Describe how the graph of can be obtained from the graph

of ( ) .

5. Write an equation for the quadratic function whose graph

contains the vertex (2, 3) and the point (0,3).

x x

f

g x x

Quick Review Solutions 2

2 2

1. Find the distance between ( 1,2) and (3, 4).

2. Solve for in terms of . 2 6

3. Complete the square to rewrite the equation in vertex form.

2 5

4. Find

52

3

( 1

the ver

4

tex

)

y x y x

y xy

y

x x

x

2

2

and axis of the graph of ( ) 2( 1) 3.

Describe how the graph of can be obtained from the graph

vertex:( 1,3); axis: 1; translation left 1 unit,

vertical stretch by a factor of

of ( ) .

2,

f x x

f

g xx x

2

5. Write an equation for the quadratic function whose graph

contains the vertex (2, 3) and

translation up 3 u

the point (0,3).

nits.

32 3

2y x

What you’ll learn about• Conic Sections• Geometry of a Parabola• Translations of Parabolas• Reflective Property of a Parabola

… and whyConic sections are the paths of nature: Any free-

moving object in a gravitational field follows the path of a conic section.

ParabolaA parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.

A Right Circular Cone (of two nappes)

Conic Sections and Degenerate Conic

Sections

Conic Sections and Degenerate Conic Sections (cont’d)

Second-Degree (Quadratic) Equations

in Two Variables

2 2 0, where , , and , are not all zero.Ax Bxy Cy Dx Ey F A B C

Structure of a Parabola

Graphs of x2=4py

Parabolas with Vertex (0,0)

• Standard equation x2 = 4py y2 = 4px• Opens Upward or To the Right

Downword or to the left

• Focus (0,p) (p,0)• Directrix y = -p x = -p• Axis y-axis x-axis• Focal length p p• Focal width |4p| |4p|

Graphs of y2 = 4px

Example Finding an Equation of a

Parabola

Find an equation in standard form for the parabola whose directrix

is the line 3 and whose focus is the point ( 3,0).x

Example Finding an Equation of a

Parabola Find an equation in standard form for the parabola whose directrix

is the line 3 and whose focus is the point ( 3,0).x

2

2

Because the directrix is 3 and the focus is ( 3,0), the focal

length is 3 and the parabola opens to the left. The equation of

the parabola in standard from is:

4

12

x

y px

y x

Parabolas with Vertex (h,k)

• Standard equation (x-h)2 = 4p(y-k) (y-k)2 = 4p(x-h)

• Opens Upward or downward To the right or to the left

• Focus (h,k+p) (h+p,k)• Directrix y = k-p x = h-p• Axis x = h y = k• Focal length p p• Focal width |4p| |4p|

Example Finding an Equation of a

Parabola

Find the standard form of the equation for the parabola with

vertex at (1,2) and focus at (1, 2).

Example Finding an Equation of a

Parabola Find the standard form of the equation for the parabola with

vertex at (1,2) and focus at (1, 2).

2

2

The parabola is opening downward so the equation has the form

( ) 4 ( ).

( , ) (1,2) and the distance between the vertex and the focus is

4. Thus, the equation is ( 1) 16( 2).

x h p y k

h k

p x y