Unit 6.3

1. 6.3
Parametric
Equations and
Motion
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Parametric Equations
 Parametric Curves
 Eliminating the Parameter
 Lines and Line Segments
 Simulating Motion with a Grapher
… and why
These topics can be used to model the path of an object
such as a baseball or golf ball.
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3. Parametric Curve, Parametric
Equations
The graph of the ordered pairs (x,y) where
x = f(t) and y = g(t)
are functions defined on an interval I of t-values
is a parametric curve. The equations are
parametric equations for the curve, the variable
t is a parameter, and I is the parameter
interval.
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4. Example Graphing Parametric
Equations
For the given parametric interval, graph the
parametric equations x  t 2  2, y  3t.
(a)  3  t  1 (b)  2  t  3 (c)  3  t  3
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5. Example Graphing Parametric
Equations
For the given parametric interval, graph the
parametric equations x  t 2  2, y  3t.
(a)  3  t  1 (b)  2  t  3 (c)  3  t  3
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6. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  t 1, y  2t,    t  .
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7. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  t 1, y  2t,    t  .
Solve one equation for t:
x  t 1
t  x 1
Substitute t into the second equation:
y  2t  2(x 1)
y  2x  2
The graph of y  2x  2 is a line.
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8. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  3cost, y  3sint, 0  t  2 .
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9. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  3cost, y  3sint, 0  t  2 .
x2  y2  9cos2 t  9sin2 t
 9cos2 t  sin2 t
 9(1)
The graph of x2  y2  9 is a circle with the
center at (0,0) and a radius of 3.
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10. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  5 t 2 , y  6t.
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11. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  5 t 2 , y  6t.
Solve the second equation for t: t 
y
6
Substitute that result into the first equation:
x  5  t 2
x  5 
y
6

 
  2

x  5 
y2
36
The graph of this
equation is a parabola
that opens to the left
with vertex 5,0.
36x  180  y2
y2  180  36x
y2  36x  5
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12. Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x  5 t 2 , y  6t.
Confirm Graphically
This is consistent with
the graph.
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13. Example Finding Parametric
Equations for a Line
Find a parametrization of the line through the points
A  (2,3) and B  (3,6).
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14. Example Finding Parametric
Equations for a Line
Find a parametrization of the line through the points
A  (2,3) and B  (3,6).
Let P(x, y) be an arbitrary point on the line through A and B.
Vector OP is the tail-to-head vector sum of OA and AP.
AP is a scalar multiple of AB. Let the scalar be t and
OP  OA AP
OP  OA t  AB
x, y  2,3  t 3 2,6  3
x, y  2,3  t 5,3
x, y  2  5t,3 3t
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15. Quick Review
1. Find the component form of the vectors
(a) OA, (b) OB, and (c) AB where O is the origin,
A  (3,2) and B  (-4,-6).
2. Write an equation in point-slope form for the line
through the points (3,2) and (-4,-6).
3. Find the two functions defined implicitly by y2  2x.
4. Find the equation for the circle with the center at (2,3)
and a radius of 3.
5. A wheel with radius 12 in spins at the rate 400 rpm.
Find the angular velocity in radians per second.
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16. Quick Review Solutions
1. Find the component form of the vectors
(a) OA, (b) OB, and (c) AB where O is the origin,
A  (3,2) and B  (4, 6).
(a) 3,2 (b) 4, 6 (c) 7, 8
2. Write an equation in point-slope form for the line
through the points (3,2) and (  4,  6). y  2 
8
7
(x  3)
3. Find the two functions defined implicitly by y2  2x.
y  2x; y   2x
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17. Quick Review Solutions
4. Find the equation for the circle with the center at
(2,3) and a radius of 3.
x  22
 y  32
 9
5. A wheel with radius 12 in spins at the rate 400 rpm.
Find the angular velocity in radians per second.
40 / 3 rad/sec
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