6.3
Parametric
Equations and
Motion
Copyright © 2011 Pearson, Inc.
What you’ll learn about
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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.
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 2
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.
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 3
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
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 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
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 5
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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Slide 6.3 - 6
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.
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 7
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x = 3cost, y = 3sint, 0 £ t < 2p .
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Slide 6.3 - 8
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x = 3cost, y = 3sint, 0 £ t < 2p .
x 2 + y 2 = 9cos 2 t + 9sin 2 t
(
= 9 cos 2 t + sin 2 t
)
= 9(1)
The graph of x 2 + y 2 = 9 is a circle with the
center at (0,0) and a radius of 3.
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 9
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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Slide 6.3 - 10
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x = 5 - t 2 , y = 6t.
y
Solve the second equation for t: t =
6
Substitute that result into the first equation:
x = 5 - t2
æ yö
x = 5- ç ÷
è 6ø
2
y
x = 536
2
Copyright © 2011 Pearson, Inc.
36x = 180 - y 2
y 2 = 180 - 36x
(
The graph of this
)
y 2 = -36 x - 5
equation is a parabola
that opens to the left
( )
with vertex 5,0 .
Slide 6.3 - 11
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.
Copyright © 2011 Pearson, Inc.
Slide 6.3 - 12
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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Slide 6.3 - 13
Example Finding Parametric
Equations for a Line
Find a parametrization of the line through the points
A = (2,3) and B = (-3,6).
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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x = 2 - 5t, y = 3 + 3t
Slide 6.3 - 14
Quick Review
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Slide 6.3 - 15
Quick Review Solutions
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Slide 6.3 - 16
Quick Review Solutions
4. Find the equation for the circle with the center at
(2,3) and a radius of 3.
( x - 2) + ( y - 3)
2
2
=9
5. A wheel with radius 12 in spins at the rate 400 rpm.
Find the angular velocity in radians per second.
40p / 3 rad/sec
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Slide 6.3 - 17