Unit 9 Parametric Equations and Vector Valued-Functions
Name:
,/"/ e,/i
AP Calculus BC
Period:
Date:
/ 7
/
Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the
form y=f (x) because C fails the Vertical Line Test.
y
So, we need an alternative method for describing curve algebraically.
** Parametric Equation
If f and g are continuous functions of # on an interval 7, then the equations
= f (t) and y = g(t) are called parametric equations and 7 is called a parameter,
usually measuring time. The set of ordered pairs (x, y) obtained as ¢ varies over the interval /
is called the graph of the parametric equations. Taken together, the parametric equations and the graph are called a
plane curve, denoted by C .
** Eliminating the Parameter
Finding a rectangular equation that represents the graph of a set of parametric equations is called eliminating the
parameter. Sometimes it is desirable to eliminate the parameter to recognize the curve.
Parametric
Equations
—
Solve for ¢ in
one equation
——
Substitute into
second equation
—_
Rectangular
Equation
Example 1 Sketch the parametric curves and identify those which definey as a function of X. In each case, eliminate
the parameter and identify the curve.
A. x=\/;,y=l—2,t6[0,4],xe[—3,3],ye[—3,3]
AL
\A
¥=¢
‘
(2, 2
@
B. x=cost, y=sint, 1 €[0,27),x[-3,3],ye[-2,2]
(0./)
2
>y
fi/ =
s
.
= e
5‘//7'\{
(/,o)
(s, 0
X Y= cost + it =/
(o)1)
e =0,
: 4’7y)=’ /
{: g/
Wéfl
(0, 0)
(‘!n;{v
yedive = /
X=toe0=/,
J=smo=o0c
f:(fl”g:dl
/:9}":{:/
Crrele
N
C. x=3cost, y =sint, 16[0,27[), xe[—4,4],ys[—3,3]
X _.
Lo
i ot
"
/ 214
2
L)zt , Yoit
fl: +-/’: CohA = /
7
3>
7®
MINEES, ¥=3t0T=0, YepZ-
g‘/fi/“
)
e
= 7],
X = 27= 3, ;:S/AW;O
wWhe- f:?g/
9/:356”%7: 2, !:S/’i)%.”‘: -
L—
il
Cody (0,0), Yertwer (2,0), (~3.0)
CVepee (0,7), (0, )
D. x=sect, y=tant, te[O,Z/r), xe[—6,6],ye[—6,6]
v= sect, JEbant | Mo d=0, v=seco =/
=
2y
2
ca’t — ot
" . =7
=/ j-
Km IS
4’2
e
o
2
% =/
fé”/”j'/’?
J=bmo=o
‘
[t = cect
=
—t
2
F=cZ=
3., ;ffi’w;f@
@
@
; .5:7/{7:
-
-
&
@)
et
57 geese()3
.
W
Cotr(s,0)
:é,‘(i'l’)r'/
- gjaf:;«;rg):»v:
4/%!:5”
7 P ()
Uerteea (1,0), (7 10) <t Attt : Y=2y
>
=
(51:
Q(;) z
= AZ)=%
** Parametric Forms of Rectangular Equations
A. Line through (x,,,) and (x,,,)
Ao ek form
y=n+t(n-x)
A
(fl"‘%)i (fi”—l) = l"‘
B. Circle: Center (h,k) radius =7
[X=h+,cos,,
V-A=ret , fA4=rene
y=k+rsinf
%'/ = ol
2‘;_4 =Srhg
"(1;_4): (fi;f);
=
l(pufl—f S fi
4’—42& ('7*!) =/
C. Ellipse: Center (h,k), a>b
x=h+acosf
(1/»%) + (y»/) =)
()c—h)Z
[y=k+bsin9®
w7
a
(yfk)Z
=
b
x=h+bcos
=1
wot Sl /)
y=k+asin€®
(x—h)
(y~k)
i R
S |
b
*
a’
L/,:/
¥=4 _
,
w0+ ar’b=/
@, (A
%’
>
cecO— p= /
D. Hyperbola: Center (h,k)
x=h+asec
(xfh)2
(y*k)2
y=k+btanf
a
b’
=1
> ?J
Q’—j,;etfi fl'é r///f
secp—Fal O/
C(wh) (94)
o/
x=h+btand _ (y—k) (x—h)
[y=k+asecfl
N
a
b
&)
Example 2 Find a parametric representation for the line segment through (—2 5) and (4 13)
s%
fi(
sec 4/
7
At (%,
4) ke (=2, 5)
Wy N
¥=—2+t{4-02] [H=—2+ s
J- b+ t[-n-4] 0 | Y=g e
(%) -)+ {(’//)
o L, y)= -2, 5P+ ('((,'//?
e
Example 3 Find a parametric representation of the circle (x — 2)Z + (y + 3)2 =25
w=) , ()_
/P(Wf)—#(;TS f)S /
,._;—+/,/
2
2
#3?
4"»3
"
G2
I3
,'
e
%
A
Az =
(22)% (42
%
/
b
.
Y=t iwt
|
94.5 = & Sint
2
/s 2+ £ St
2
Example 4 Find a parametric representation
%
of
+ % =1
M + C&?f:/
'ot
{%‘/fe
\TIGETar el
sy (e p
T (wt)romt)
y-f/: wt
>
—
{Q,/; —/+ (oot
Ve,
’4/: spnt
Zi
/7
Y (5,4)
[2&7Z0
Example 5 Find a parametric representation of
2
3%
X= =3+ 3lont
/f
=
§-€C 2‘/ -
&%){ = /
M- tat
oot
J2
2
72
21 zgcect