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10.6 Parametric Equations
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Definition of Parametric Equations
parametric equation is a method of defining a relation using parameters. A simple kinematic1 example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion.
1. Kinematic.The branch of mechanics that studies the motion of a body or a system of bodies without consideration given to its mass or the forces acting on it.
http://en.wikipedia.org/wiki/Parametric_equation
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We will not be finding parametric equations, in this example.
In this class we will graph parametric equation also eliminating the parameter.
Given the equation With respect to time
Rectangular equation Parametric Equations
The parametric equations have to be Continuous
xx
y
72
2
tty
tx
22416
2242
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Both Equations graph the same. The Parametric states how fast to graph
Given the equation With respect to time
Rectangular equation Parametric Equations
xx
y
72
2
tty
tx
22416
2242
0,72
2
23t
4
23t
18,36
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Graphing Parameter equations
Parameter equations
2
4
1
ty
tx
t x y-4 -1 16-2 -0.5 40 0 02 0.5 44 1 16
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Graphing Parameter equations
Parameter equations
2
4
1
ty
tx
t x y-4 -1 16-2 -0.5 40 0 02 0.5 44 1 16
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Change to Rectangular Equation
Parameter equations to
Rectangular equation
t x y-4 -1 16-2 -0.5 40 0 02 0.5 44 1 16
2
22
16
4
44
1
xy
xyty
txtx
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Change from Parametric to Rectangular
Parametric Equations
121 xxtxx 121 yytyy
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Change from Parametric to Rectangular
Parametric Equations
txx
xx
xxtxx
12
1
121 121 yytyy
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Change from Parametric to Rectangular
Parametric Equations
txx
xx
xxtxx
12
1
121
12
12
11
121
yyxx
xxyy
yytyy
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Change from Parametric to Rectangular
Parametric Equations
txx
xx
xxtxx
12
1
121
12
1211
12
12
11
121
xx
yyxxyy
yyxx
xxyy
yytyy
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Change from Parametric to Rectangular
Parametric Equations
txx
xx
xxtxx
12
1
121
mxxyy
xx
yyxxyy
yytyy
11
12
1211
121
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Change from Parametric to Rectangular
Parametric Equations
txx
xx
xxtxx
12
1
121
11
12
1211
121
xxmyy
xx
yyxxyy
yytyy
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Change from Parametric to Rectangular
x = h + a Cosθ y = k + b Sinθ
Cosa
hx
Sinb
ky
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Change from Parametric to Rectangular
x = h + a Cosθ y = k + b Sinθ
Cosa
hx
Sinb
ky
1
1
1
2
2
2
2
22
22
b
ky
a
hx
b
ky
a
hx
SinCosSince
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What would the parametric equation be for a circle with center at (3,2) and radius of 4
(x – h)2 + (y – k)2 = r2 In a circle a = b = 4
Cosr
hx
Sinr
ky
1
2
2
2
2
r
ky
r
hx
SinySiny
CosxCosx
424
2
434
3
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Homework
Page 747 – 749
# 7, 14, 21, 28,
32, 41, 48
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Homework
Page 747 – 749
# 5, 10, 15, 20,
25, 30, 35, 40