What’s in a ParabolaA parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point called the focus.

ParabolaThe Standard Form of a Parabola that opens to the right and has a vertex at (0,0) is……

pxy 42

ParabolaThe Parabola that opens to the right and has a vertex at (0,0) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (p,0)This makes the equation of the directrix x = -pThe makes the axis of symmetry the x-axis (y = 0)

ParabolaThe Standard Form of a Parabola that opens to the left and has a vertex at (0,0) is……

pxy 42

ParabolaThe Parabola that opens to the left and has a vertex at (0,0) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus(-p,0)This makes the equation of the directrix x = pThe makes the axis of symmetry the x-axis (y = 0)

ParabolaThe Standard Form of a Parabola that opens up and has a vertex at (0,0) is……

pyx 42

ParabolaThe Parabola that opens up and has a vertex at (0,0) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (0,p)This makes the equation of the directrix y = -pThis makes the axis of symmetry the y-axis (x = 0)

ParabolaThe Standard Form of a Parabola that opens down and has a vertex at (0,0) is……

pyx 42

©1999 Addison Wesley Longman, Inc.

ParabolaThe Parabola that opens down and has a vertex at (0,0) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (0,-p)This makes the equation of the directrix y = pThis makes the axis of symmetry the y-axis (x = 0)

ParabolaThe Standard Form of a Parabola that opens to the right and has a vertex at (h,k) is……

)(4)( 2 hxpky

ParabolaThe Parabola that opens to the right and has a vertex at (h,k) has the following characteristics……..

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (h+p, k)This makes the equation of the directrix x = h – pThis makes the axis of symmetry

aby

2

ParabolaThe Standard Form of a Parabola that opens to the left and has a vertex at (h,k) is……

)(4)( 2 hxpky

ParabolaThe Parabola that opens to the left and has a vertex at (h,k) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (h – p, k)This makes the equation of the directrix x = h + pThe makes the axis of symmetry

aby

2

ParabolaThe Standard Form of a Parabola that opens up and has a vertex at (h,k) is……

)(4)( 2 kyphx

ParabolaThe Parabola that opens up and has a vertex at (h,k) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (h , k + p)This makes the equation of the directrix y = k – p

The makes the axis of symmetry

abx

2

ParabolaThe Standard Form of a Parabola that opens down and has a vertex at (h,k) is……

)(4)( 2 kyphx

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

ParabolaThe Parabola that opens down and has a vertex at (h,k) has the following characteristics……

p is the distance from the vertex of the parabola to the focus or directrixThis makes the coordinates of the focus (h , k - p)This makes the equation of the directrix y = k + p

This makes the axis of symmetry

abx

2

Hyperbola

The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

© Jill Britton, September 25, 2003

What is a Hyperbola?The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

Where are the Hyperbolas?A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.

Hyperbola General Rules

x and y are both squared Equation always equals(=) 1 Equation is always minus(-) a2 is always the first denominator c2 = a2 + b2 c is the distance from the center to each foci on

the major axis a is the distance from the center to each vertex on

the major axis

Hyperbola General Rules

b is the distance from the center to each midpoint of the rectangle used to draw the asymptotes. This distance runs perpendicular to the distance (a).

Major axis has a length of 2a Eccentricity(e): e = c/a (The closer e gets to 1,

the closer it is to being circular If x2 is first then the hyperbola is horizontal If y2 is first then the hyperbola is vertical.

Hyperbola General Rules

The center is in the middle of the 2 vertices and the 2 foci.

The vertices and the covertices are used to draw the rectangles that form the asymptotes.

The vertices and the covertices are the midpoints of the rectangle

The covertices are not labeled on the hyperbola because they are not actually part of the graph

Hyperbola The standard form of the Hyperbola with a center at

(0,0) and a horizontal axis is……

12

2

2

2

by

ax

Hyperbola The Hyperbola with a center at (0,0) and a horizontal

axis has the following characteristics……

Vertices ( a,0) Foci ( c,0)

Asymptotes: xaby

Hyperbola The standard form of the Hyperbola with a center at

(0,0) and a vertical axis is……

12

2

2

2

bx

ay

Hyperbola The Hyperbola with a center at (0,0) and a vertical

axis has the following characteristics……

Vertices (0, a) Foci ( 0, c)

Asymptotes: x

bay

Hyperbola The standard form of the Hyperbola with a center at

(h,k) and a horizontal axis is……

1)()(2

2

2

2

bky

ahx

Hyperbola The Hyperbola with a center at (h,k) and a horizontal

axis has the following characteristics……

Vertices (h a, k) Foci (h c, k )

Asymptotes:

)( hxabky

Hyperbola The standard form of the Hyperbola with a center at

(h,k) and a vertical axis is……

1)()(2

2

2

2

bhx

aky

Hyperbola The Hyperbola with a center at (h,k) and a vertical

axis has the following characteristics……

Vertices (h, k a) Foci (h, k c)

Asymptotes: )( hx

baky

Rotating the Coordinate Axis

022 FEyDxCyBxyAx

Equations for Rotating the Coordinate Axes

sin'cos' yxx

cos'sin' yxy

BCA

2cot CAB

2tanor

PARABOLAS

DefinitionThe set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix.

3.5

3

2.5

2

1.5

DIRECTRIX

FOCUS Same Distance!POINT

Writing linear equation in parabolic form

GOAL: Turn 2 into y ax bx c

2( )y a x h k

Writing linear equation in parabolic form

1. Start with 2. Group the two x-terms3. Pull out the constant with x2 from the grouping4. Complete the square of the grouping

**Look back to Topic 6.3 for help**

5. Write the squared term as subtraction so that you end with

2 y ax bx c

2( ) y a x h k

**Remember that whatever you add in the grouping must be subtracted from the c-value**

Group x-termsPull out GCF

Complete the Square

Factor and simplify

4))2((3 2 xy

or 4)2(3 2 xy

1216)44(3 2 xxy

____16___)4(3 2 xxy16)4(3 2 xxy16)123( 2 xxy

16123: 2 xxyExample

Why write in parabolic form?

It gives you necessary information to draw the parabola

1 ,4

h ka

EquationAxis of symmetry x = h y = kVertex (h, k) (h, k)Focus

Directrix

Direction of opening

Up: a>0, Down: a<0

Right: a>0, Left: a<0

Latus Rectum

2( )y a x h k 2( )x a y k h

1,4

h ka

14

y ka

1

4x h

a

1 unitsa

1 unitsa

Graph of prior example

23( ( 2)) 4 y x

length of latus rectum: 1

12

directrix: y = 47

12

axis of symmetry: x = -2

vertex: (-2,4) focus: (-2,49

12)

You Try!!

Write the following equation in parabolic form. State the vertex, axis of symmetry and direction of opening.2 10 7x y y

2Parabolic form: ( 5) 32x y Vertex: (5, -32)Axis of symmetry: 5x Direction of Opening: upward

Mechanical Engineering Dept. 43

Conic Curves - Parabolas

Ken Youssefi

Conic curves or conics are the curves formed by the intersection of a plane with a right circular cone (parabola, hyperbola and sphere).A parabola is the curve created when a plane intersects a right circular cone parallel to the side (elements) of the cone

Cutting plane

Parallel

Mechanical Engineering Dept. 44

Conic Curves - Hyperbolas

Ken Youssefi

A hyperbola is the curve created when a plane parallel to the axis and perpendicular to the base intersects a right circular cone.

Element (side)

Hyperbola

Orthographic view

A hyperbola is a set of points in a plane the difference of whose distances from two fixed points, called foci, is a constant.

F1 F2

d1d2

PFor any point P that is on the hyperbola, d2 – d1 is always the same.

In this example, the origin is the center of the hyperbola. It is midway between the foci.

F F

V V

C

A line through the foci intersects the hyperbola at two points, called the vertices.

The segment connecting the vertices is called the transverse axis of the hyperbola.The center of the hyperbola is located at the midpoint of the transverse axis.

As x and y get larger the branches of the hyperbola approach a pair of intersecting lines called the asymptotes of the hyperbola. These asymptotes pass through the center of the hyperbola.

F

F

V

V

C

The figure at the left is an example of a hyperbola whose branches open up and down instead of right and left.

Since the transverse axis is vertical, this type of hyperbola is often referred to as a vertical hyperbola.

When the transverse axis is horizontal, the hyperbola is referred to as a horizontal hyperbola.

(x – h)2 (y – k)2

a2 b2

Horizontal Hyperbola

(y – k)2 (x – h)2

b2 a2– = 1

Vertical Hyperbola

– = 1

The center of a hyperbola is at the point (h, k) in either form

For either hyperbola, c2 = a2 + b2

Where c is the distance from the center to a focus point.The equations of the asymptotes are

y = (x – h) + k and y = (x – h) + kba

ba

-

A parabola is a set of points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus.

For any point Q that is on the parabola, d2 = d1

Directrix

FocusQ

d1

d2The latus rectum of a parabola is a line segment that passes through the focus, is parallel to the directrix and has its endpoints on the parabola. The length of the latus rectum is |4p| where p is the distance from the vertex to the focus.

V

Things you should already know about a parabola.

Forms of equationsy = a(x – h)2 + k

opens up if a is positiveopens down if a is negativevertex is (h, k)

y = ax2 + bx + copens up if a is positiveopens down if a is negativevertex is , f( )-b

2a-b 2a

Thus far in this course we have studied parabolas that are vertical - that is, they open up or down and the axis of symmetry is vertical

In this unit we will also study parabolas that are horizontal – that is, they open right or left and the axis of symmetry is horizontal

In these equations it is the y-variable that is squared.

V

x = a(y – k)2 + h

x = ay2 + by + c

or

Horizontal Hyperbola

Vertical Hyperbola

If a > 0, opens rightIf a < 0, opens left

The directrix is vertical

x = ay2 + by + c y = ax2 + bx + c

Vertex: x = If a > 0, opens upIf a < 0, opens down

The directrix is horizontal

Remember: |p| is the distance from the vertex to the focus

vertex: -b 2ay = -b

2a

a = 14p

the directrix is the same distance from the vertex as the focus is

A hyperbola is created from the intersection of a plane with a double cone.

A hyperbola is defined by a group of points that have a same difference of distance from two foci.

When you subtract the small line from the long line for each ordered pair the remaining value is the same.

Hyperbolas can be symmetrical around the x-axis or the y-axis The one on the right is symmetrical around the x-axis.

A hyperbola is a set of points in a plane the difference of whose distances from two fixed points, called foci, is a constant.

F1 F2

d1d2

PFor any point P that is on the hyperbola, d2 – d1 is always the same.

In this example, the origin is the center of the hyperbola. It is midway between the foci.

F F

V V

C

A line through the foci intersects the hyperbola at two points, called the vertices.

The segment connecting the vertices is called the transverse axis of the hyperbola.The center of the hyperbola is located at the midpoint of the transverse axis.

As x and y get larger the branches of the hyperbola approach a pair of intersecting lines called the asymptotes of the hyperbola. These asymptotes pass through the center of the hyperbola.

F

F

V

V

C

The figure at the left is an example of a hyperbola whose branches open up and down instead of right and left.

Since the transverse axis is vertical, this type of hyperbola is often referred to as a vertical hyperbola.

When the transverse axis is horizontal, the hyperbola is referred to as a horizontal hyperbola.