Parametric Equations
10.6
Adapted by JMerrill,
2011
Plane Curves
• Up to now, we have been representing
graphs by a single equation in 2 variables.
The y = equations tell us where an object
(ball being thrown) has been.
• Now we will introduce a 3rd variable, t
(time) which is the parameter. It tells us
when an object was at a given point on a
path.
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2
The path of an object thrown into the air at a 45°
angle at 48 feet per second can be represented by
2
x
y    x. Rectangular equation
72
horizontal distance (x)
vertical distance (y)
A pair of parametric equations are equations with
both x and y written as functions of time, t.
Now the distances
depend on the
time, t.
Parametric equation for x
x  24 2t
y  16t 2  24 2t Parametric equation for y
t is the parameter.
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3
Example:
2
x
y x
72
y
18
Parametric
equations
x  24 2t
y  16t 2  24 2t
t3 2
4
(36, 18)
9
t3 2
2
(72, 0)
x
9
18
27 36
45
54
63
72
(0, 0)
t=0
two variables (x and y) for position
Curvilinear motion:
one variable (t) for time
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4
Example:
Sketch the curve given by
x = t + 2 and y = t2, – 3  t  3.
t –3 –2 –1
x –1 0 1
0
2
1
3
2
4
3
5
y
0
1
4
9
9
4
1
y
8
4
The (x,y) ordered pairs will
graph exactly the same as
they always have graphed.
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x
-4
4
5
Graphing Utility: Sketch the curve given by
x = t + 2 and y = t2, – 3  t  3.
Mode Menu:
Set to parametric mode.
Window
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Graph
Table
6
Eliminating the parameter is a process for finding
the rectangular equation (y =) of a curve represented
by parametric equations.
x=t+2
y = t2
Parametric equations
t=x–2
Solve for t in one equation.
y = (x –2)2
Substitute into the second equation.
y = (x –2)2
Equation of a parabola with the vertex at (2, 0)
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7
Example:
Identify the curve represented by x = 2t and y  t  2
by eliminating the parameter.
tx
Solve for t in one equation.
2
y  x  2 Substitute into the second equation.
2
y
The absolute value bars
can be found in the Math
menu--Num
8
4
y  x 2
2
x
-4
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4
8
Eliminating an Angle Parameter
• Sketch and identify the curve represented
by x = 3cosθ, y = 4sinθ
y
x
cos


sin


• Solve for cosθ & sinθ:
3
4
• Use the identity cos2θ + sin2θ = 1
x  y  1
   
3 4
x2 y2

1
9
16
2
• We have a vertical ellipse
with a = 4 and b = 3
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2
9
You Try
• Eliminate the parameter in the equations
x  3  2t
y  2  3t
2t  x  3
x 3
t 

2
2
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x 3 

y  2  3
 
2
 2
3
9
y  2
x 
2
2
3
11
y 
x 
2
2
10
Writing Parametric Equations from Rectangular
Equations
Find a set of parametric equations to represent the
graph of y = 4x – 3.
x=t
Let x = t
y = 4t – 3
Substitute into the original rectangular equation.
y
y = 4t – 3
8
4
4
-4
x
-4
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11
You Try
• Find a set of parametric equations given
y = x2
•
•
x=t
y = t2
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12
Application:
The center-field fence in a ballpark is 10 feet high and 400 feet
from home plate. A baseball is hit at a point 3 feet above the
ground and leaves the bat at a speed of 150 feet per second at
an angle of 15. The parametric equations for its path are
x = 145t and y = 3 + 39t – 16t2.
Graph the path of the baseball. Is the hit a home run?
y
25
The ball only traveled
364 feet and was not a
home run.
20
15
10
(364, 0)
5
(0, 3)
0 50
100
150 200 250 300 350 400
x
Home Run
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13