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Copyright © 2011 Pearson, Inc.
6.3Parametric
Equations and Motion
Slide 6.3 - 2 Copyright © 2011 Pearson, Inc.
What you’ll learn about
Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher
… and whyThese topics can be used to model the path of an object such as a baseball or golf ball.
Slide 6.3 - 3 Copyright © 2011 Pearson, Inc.
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.
Slide 6.3 - 4 Copyright © 2011 Pearson, Inc.
Example Graphing Parametric Equations
For the given parametric interval, graph the
parametric equations x =t2 −2, y=3t.(a) −3≤t≤1 (b) −2 ≤t≤3 (c) −3≤t≤3
Slide 6.3 - 5 Copyright © 2011 Pearson, Inc.
Example Graphing Parametric Equations
For the given parametric interval, graph the
parametric equations x =t2 −2, y=3t.(a) −3≤t≤1 (b) −2 ≤t≤3 (c) −3≤t≤3
Slide 6.3 - 6 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x =t+1, y=2t, −∞ < t< ∞.
Slide 6.3 - 7 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
Solve one equation for t:
x =t−1t=x−1Substitute t into the second equation:y=2t=2(x−1)y=2x−2The graph of y=2x−2 is a line.
Eliminate the parameter and identify the graph of the
parametric curve x =t+1, y=2t, −∞ < t< ∞.
Slide 6.3 - 8 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x =3cost, y=3sint, 0 ≤t< 2π.
Slide 6.3 - 9 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
x2 + y2 =9cos2 t+9sin2 t
=9 cos2 t+sin2 t( )
=9(1)
The graph of x2 + y2 =9 is a circle with thecenter at (0,0) and a radius of 3.
Eliminate the parameter and identify the graph of the
parametric curve x =3cost, y=3sint, 0 ≤t< 2π.
Slide 6.3 - 10 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x =5−t2 , y=6t.
Slide 6.3 - 11 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
Solve the second equation for t: t =y6
Substitute that result into the first equation:
x=5−t2
x=5−y6⎛
⎝⎜⎞
⎠⎟
2
x=5−y2
36
Eliminate the parameter and identify the graph of the
parametric curve x =5−t2 , y=6t.
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( )
Slide 6.3 - 12 Copyright © 2011 Pearson, Inc.
Example Eliminating the Parameter
Eliminate the parameter and identify the graph of the
parametric curve x =5−t2 , y=6t.
Confirm Graphically
This is consistent with
the graph.
Slide 6.3 - 13 Copyright © 2011 Pearson, Inc.
Example Finding Parametric Equations for a Line
Find a parametrization of the line through the points
A =(2,3) and B=(−3,6).
Slide 6.3 - 14 Copyright © 2011 Pearson, Inc.
Example Finding Parametric Equations for a Line
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
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
Slide 6.3 - 15 Copyright © 2011 Pearson, Inc.
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 linethrough 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.
Slide 6.3 - 16 Copyright © 2011 Pearson, Inc.
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 =87(x−3)
3. Find the two functions defined implicitly by y2 =2x.
y= 2x; y=− 2x
Slide 6.3 - 17 Copyright © 2011 Pearson, Inc.
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.
40π / 3 rad/sec
