parametric equations 10.6 adapted by jmerrill, 2011
TRANSCRIPT
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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A pair of parametric equations are equations with both x and y written as functions of time, t.
216 24 2y t t Parametric equation for x
Parametric equation for y
t is the parameter.
Rectangular equation2
.72xy x
The path of an object thrown into the air at a 45° angle at 48 feet per second can be represented by
horizontal distance (x)vertical distance (y)
24 2x tNow the distances depend on the
time, t.
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216 24 2y t t 2
72xy x
y
x
9
18
9 18 27 36 45 54 63 72(0, 0)t = 0
(36, 18)
3 24
t
3 22
t (72, 0)
two variables (x and y) for positionone variable (t) for time
Curvilinear motion:
24 2x tExample:Parametric equations
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Example:Sketch the curve given by
x = t + 2 and y = t2, – 3 t 3.
t – 3 – 2 – 1 0 1 2 3
x – 1 0 1 2 3 4 5
y 9 4 1 0 1 4 9 y
x-4 4
4
8
The (x,y) ordered pairs will graph exactly the same as they always have graphed.
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Graphing Utility: Sketch the curve given by x = t + 2 and y = t2, – 3 t 3.
Mode Menu:
Set to parametric mode.
Window Graph Table
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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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Solve for t in one equation.
Substitute into the second equation.
Example:2y t Identify the curve represented by x = 2t and
by eliminating the parameter.
2xt
22y x
y
x-4 4
4
8
22xy
The absolute value bars can be found in the Math
menu--Num
Eliminating an Angle Parameter
• Sketch and identify the curve represented by x = 3cosθ, y = 4sinθ
• Solve for cosθ & sinθ:
• Use the identity cos2θ + sin2θ = 1
• We have a vertical ellipse with a = 4 and b = 3
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3 4cos sin
yx
22
22
13 4
19 16
yx
yx
You Try
• Eliminate the parameter in the equations
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3 2
2 3
x t
y t
2 3
3
2 2
t x
xt
32 3
2 23 9
22 2
3 11
2 2
xy
y x
y x
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Let x = t
Substitute into the original rectangular equation.
Writing Parametric Equations from Rectangular Equations Find a set of parametric equations to represent the graph of y = 4x – 3.
x = t
y = 4t – 3
x
y
-4 4
4
-4
8y = 4t – 3
You Try
• Find a set of parametric equations given y = x2
• x = t
• y = t2
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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?
Home Run
(364, 0)
y
5
10
0
15
20
25
x50 100 150 200 250 300 350 400
The ball only traveled 364 feet and was not a home run.
(0, 3)