Intro to Conics- Parabolas - WMwikis

9­1
Intro
to
Conics:
Parabolas



Objective:

Students
will
write
the
equation
of
a
parabola
in
standard
form
and

identify
the
focus
and
directrix




Related
SOL:
MA.8
The
student
will
investigate
and
identify
the
characteristics
of
conic
section

equations
in
(h,k)
and
standard
forms.
The
techniques
of
translation
and
rotation
of

axes
in
the
coordinate
plane
will
be
used
to
graph
conic
sections.




Materials


‐ 30
cone
shaped
paper
water
cups

‐ patty
paper




Overview:


‐ warm
up:
double
napped
cone
activity

‐ java
applet
on
conics

‐ notes:
patty
paper
activity

‐ practice
problems




Lesson



Warm
up:

(10
min)


‐ In
pairs:
give
students
a
“double
napped
cone”
–
maybe
two
cups
taped

together.

Write
down
all
the
shapes
you
get
when
you
pass
a
plane
through

the
double
napped
cone.




‐ After
5
min
or
so,
have
students
draw
their
different
shapes
on
board
–

separate
ones
that
pass
through
the
center
and
ones
that
do
not.




General
Conics

(10
min)

Run
this
clip
to
demonstrate
all
the
non‐degenerate
forms
of
a
conic.

http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/emat6690/Insunit/con
se.avi



Conic
–
a)
the
intersection
of
a
plane
and
a
double‐napped
cone.





b)
the
collection
of
points
that
satisfy
a
certain
geometric
property.

(ie,
a

circle
is
the
collection
of
points
equidistant
from
the
center.)



Kinds
of
conics:


‐ circle

‐ parabola

‐ ellipse

‐ hyperbola

Algebraic
Definition:

Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0 


!


 All
conics
can
be
represented
in
this
form
in
some
way
–
changing
the
value


of
the
coefficient
changes
the
type
of
conic
the
equation
represents.

For
example
–

what
is
a
simple
equation
for
a
parabola?

(Students
tell
me:
y=x^2)

What
about
a


circle?
(Students
tell
me
 x2 + y2 = 1).



 Note
–
this
is
a
quadratic
equation
–
you
might
think
about
 ax2 + bx + c 
being

the
definition
of
a
quadratic,
but
a
quadratic
equation
is
simply
an
equation
with
a

degree
of
2.


 ! €
Parabolas




Activating
Prior
Knowledge:
(5
min)


Already
are
familiar
with
one
algebraic
definition
of
a
parabola
–
studied
it
in
Alg.
2.


General
form
of
the
equation
for
a
parabola,
as
we
know
so
far:
(let
students
try
and


remember)


f (x) = ax 2 + bx + c 


Note:
the
parabola
can
only
open
up
or
down
according
to
this
definition.

Today


we’re
also
going
to
look
at
parabolas
that
open
to
the
left
and
right.



! Geometric
definition:
(15
min)



 Pass
out
patty
paper.

Have
students
draw
a
straight
line
close
to
the
bottom

of
the
paper.

(will
be
the
directrix.)

Then
they
should
draw
a
point
somewhere
in


the
middle
of
the
paper.

(will
be
the
focus.)

Have
them
draw
several
points
along

their
line
and
fold
the
paper
so
that
the
focus
maps
onto
each
of
those
points.

Then


they
may
continue
folding,
mapping
the
point
onto
the
line
anywhere
they
wish.



The
result
will
be
a
parabola.

From
the
activity,
derive
this
definition:


A
parabola
is
the
set
of
all
points
(x,y)
in
a
plane
that
are
equidistant
from
a
fixed

line
–
the
directrix,
and
a
point
not
on
the
line–
the
focus.















































p
is
the
distance
from
the
vertex
to
the
focus.

If
p
>0,
parabola
opens
up
or
right.

If
p


<0,
parabola
opens
down
or
left.




Standard
form
of
a
Parabola

(5
min)

‐ have
students
find
the
formulas
for
directrix
&
focus






Vertical
Axis:

 (x " h)2 = 4 p(y " k) p#0



 Directrix:

 y=k­p
 
 


Focus:


 (h,k+p)


Horizontal
Axis:
 (y " h)2 = 4 p(x " k) p#0


 Direc!trix:

x=h­p
 
 


Focus:



 (h+p,k)




Examples:
(!20
min)


(handwritten)




Reflective
Property





All
incoming
rays
parallel
to
the
axis
of
symmetry
will
be
reflected
through
the
focus


of
the
parabola.



All
rays
emanating
from
the
focus
will
reflect
back
parallel
to
the
axis
of
symmetry.





































Practice
(20
min)



‐ practice
problems
handwritten,
attached

‐ game
format


















HW:
13‐19
odd
37‐41
odd,
56