Transcript 10 2 Parabolas

Objectives:
1. To define a parabola
as a conic section and
as a locus and find its
parts
2. To write the equation
of a parabola in
standard form
3. To graph the equation
of a parabola in
standard form
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Assignment:
P. 740: 1-4
P. 740: 11-24 (Some)
P. 740: 25-28 (Some)
P. 741: 45-50 (Some)
P. 741-2: 61, 65
P. 743: 69, 70
Homework
Supplement
When a solid is cut by a plane, the resulting
plane figure is called a section. A section that
is parallel to the base is a cross-section.
Below is what is known as a double-napped
cone (aka double cone).
Each cone is
called a nappe.
Upper nappe
Vertex
The point of
intersection is
the vertex.
Lower nappe
Describe each of the following sections of the
double cone, none of which touch the vertex:
1. Plane is parallel to the “base”
2. Same plane as above except
tilted a few degrees
3. Plane is parallel to one of the
lateral sides
4. Plane is perpendicular to the
“base”
You will be able to define a
parabola as a conic section and as
a locus and find its parts (vertex,
focus, and directrix)
Here are the four basic conic sections, which are
formed by the intersection of a plane and a
double cone.
Plane does not pass through the vertex
Conics were first described by the Greeks and
were later instrumental in the development of
calculus in the 1500-somethings.
Plane does not pass through the vertex
Conics can be described by the general
quadratic equation: Ax2  By 2  Cx  Dy  E  0
And here are some degenerate conic sections.
What’s the difference? They’re the
simpletons of the conic family of sections.
Plane passes through the vertex
Like all conic sections,
there are plenty of
parabolic
applications:
• Galileo discovered
that trajectories of
launched objects
are parabolas
Like all conic sections,
there are plenty of
parabolic
applications:
• Galileo discovered
that trajectories of
launched objects
are parabolas
Like all conic sections,
there are plenty of
parabolic
applications:
• While the orbits of
some celestial
bodies are
elliptical, others are
parabolic
Like all conic
sections, there
are plenty of
parabolic
applications:
• Parabolic
reflectors have
parabolic crosssections
Incoming light,
sound, or
whatever that
arrives at a
parabolic
reflector parallel
to the axis of
symmetry is
directed to the
focus.
Likewise, energy
that is emitted
from the focus of
a parabolic
reflector and
then strikes the
parabola is
directed parallel
to the axis of
symmetry.
A locus is a set of points that share a common
geometric property.
A circle is the locus
of coplanar
points (x, y) that  x  h    y  k   r
are equidistant
 x  h   y  k   r
from a given
point called the
center.
2
2
2
2
2
Parabolas can be defined as a function of x:
f ( x)  a  x  h   k
2
• This parabola either
opens up or down
depending on the value
of a
• What about all of those
parabolas that open to
the left or right?
Parabolas can also be defined as a locus:
• A parabola is the set of
coplanar points (x, y)
that are equidistant
from a fixed line called a
directrix and a fixed
point not on the line
called the focus.
Click to watch me move!
The vertex (h, k) is the
midpoint between the
focus and the directrix.
• The vertex lies on the
axis of symmetry
• The distance from the
vertex to the focus is p,
the focal length
If the directrix is vertical,
then the parabola opens
to the left or right.
Vertical Directrix
Vertex
(h, k)
Focus
(h + p, k)
Directrix
x=h–p
Axis of
Symmetry
y=k
The vertex of a parabola has coordinates (2, 5)
while the focus is at (4, 5).
1. What is the focal length?
2. Is the axis of symmetry vertical or horizontal?
3. Is the directrix vertical or horizontal?
4. Does the parabola open up/down or
right/left?
You will be able to
write the equation
of a parabola in
standard form
Use the locus definition
of a parabola to
derive the equation
of a parabola with a
vertical directrix, a
vertex at (h, k), and a
focal length of p.
According to the definition:
𝑃𝐴 = 𝑃𝐹
(h – p, y)
𝑥− ℎ−𝑝
2
+ 𝑦−𝑦
2
=
𝑥− ℎ+𝑝
2
+ 𝑦−𝑘
2
According to the definition:
𝑃𝐴 = 𝑃𝐹
2
=
2
= 𝑥− ℎ+𝑝
𝑥 2 − 2𝑥 ℎ − 𝑝 + ℎ − 𝑝
2
= 𝑥 2 − 2𝑥 ℎ + 𝑝 + ℎ + 𝑝
−2𝑥ℎ + 2𝑥𝑝 + ℎ − 𝑝
2
= −2𝑥ℎ − 2𝑥𝑝 + ℎ + 𝑝
4𝑥𝑝 + ℎ − 𝑝
2
= ℎ+𝑝
𝑥− ℎ−𝑝
2
+ 𝑦−𝑦
𝑥− ℎ−𝑝
2
𝑥− ℎ+𝑝
2
2
+ 𝑦−𝑘
+ 𝑦−𝑘
2
2
4𝑝 𝑥 − ℎ = 𝑦 − 𝑘
2
Focal length
Coordinates of the Vertex
2
2
+ 𝑦−𝑘
+ 𝑦−𝑘
2
4𝑥𝑝 + ℎ2 − 2ℎ𝑝 + 𝑝2 = ℎ2 + 2ℎ𝑝 + 𝑝2 + 𝑦 − 𝑘
4𝑥𝑝 − 4ℎ𝑝 = 𝑦 − 𝑘
2
+ 𝑦−𝑘
2
2
2
If the directrix is vertical,
then the parabola opens
to the left or right.
( y – k)2 = 4 p(x – h)
Vertex
(h, k)
Focus
(h + p, k)
Directrix
x=h–p
Axis of
Symmetry
y=k
Write the equation of the parabola whose
vertex is (0, 0) and whose focus is (p, 0).
If the directrix is horizontal,
then the parabola opens
up or down.
(x – h)2 = 4 p( y – k)
Vertex
(h, k)
Focus
(h, k + p)
Directrix
y=k–p
Axis of
Symmetry
x=h
Write the equation of the parabola whose
vertex is (0, 0) and whose focus is (0, p).
What is the relationship between a and p?
y  a  x  h  k
2
 x  h
2
 4p y  k
Find the equation of the parabola in standard
form with its vertex at the origin and focus at
(3, 0).
Find the equation of the parabola in standard
form with its vertex at (3, 2) and focus at
(1, 2).
Consider the relationship between the vertex
and focus (which lies on the axis of symmetry)
and the directrix (which is perpendicular to
the axis of symmetry).
 x  h  4 p  y  k 
2
Vertex
Focus
(3, 2)
(3, 5)
Vertical axis of symmetry
Consider the relationship between the vertex
and focus (which lies on the axis of symmetry)
and the directrix (which is perpendicular to
the axis of symmetry).
 y  h  4 p  x  k 
2
Vertex
Focus
(3, 2)
(5, 2)
Horizontal axis of symmetry
Consider the two quadratic equations below.
x 2  10 x  8 y  17  0
y2  2x  6 y  7  0
1. How can we tell that these equations
represent parabolas?
2. How can we tell if the parabola will open
up/down or left/right?
Consider the two quadratic equations below.
x 2  10 x  8 y  17  0
y2  2x  6 y  7  0
Two zeros on the x-axis
Two zeros on the y-axis
Find the vertex,
focus, and
directrix of the
parabola given by
x 2  10 x  8 y  17  0
Find the vertex,
focus, and
directrix of the
parabola given by
y2  2x  6 y  7  0
You will be able to graph the
equation of a parabola in
standard form
The latus rectum of a
parabola is the
segment that passes
through the focus, is
parallel to the
directrix, and has
endpoints that are on
the parabola.
Find the length of the latus rectum of x2 = 4py.
Let the latus rectum help you graph a parabola.
Since its length is 4p, it will be ±2p from the
focus.
 x  h
2
 4p y  k
Length of latus rectum
1. Plot vertex and focus
2. Plot endpoints of latus rectum
3. Draw parabola
Graph the equation
y2  4x  4 y  4  0
Objectives:
1. To define a parabola
as a conic section and
as a locus and find its
parts
2. To write the equation
of a parabola in
standard form
3. To graph the equation
of a parabola in
standard form
•
•
•
•
•
•
•
Assignment:
P. 740: 1-4
P. 740: 11-24 (Some)
P. 740: 25-28 (Some)
P. 741: 45-50 (Some)
P. 741-2: 61, 65
P. 743: 69, 70
Homework
Supplement