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Parameterization
Section 1
Parametrically Defined Curves
So far, we dealt with relations of the form y = f(x) or F(x,y) = 0
such as: y = 5x or x2 + y2 – 4 = 0, which state a direct relationship between the
variables x and y. However, sometimes, it is more useful to express both x and y in terms of a third variable, which we will call a parameter.
Orientation
Let C be a curve defined by the equations:
x = f(t) , y = g(t) ; a ≤ t ≤ b
The direction of the increasing of the parameter t is called the orientation imposed on C by the parameter.
Examples
Example 1
Let:x = t , y = 2t ; 0 ≤ t
To graph this curve, consider the following table txy
000
112
224
336
Plot the points indicated in the table.Join these points. What do you get?
Eliminating the parameter
Now, let's examine the situation differently by eliminating the parameter t. Doing that, we get:
y = 2x; xε[0,∞) How?
The curve defined by this equation is the line segment situated on the first quadrant of the straight line through the origin and the point ( 1 ,2 ).
Example 2
Graph the curve defined by the parametric equations: x = t + 1 , y = t2 + 4t + 6
txy
-6-518
-5-411
-4-36
-3-23
-2-12
-103
016
1211
2318
Plot the points indicated in the table.
Join these points. What do you get?
Eliminating the parameter
we have:t = x – 1And soy = (x-1)2 + 4(x-1) + 6 = x2 + 2x + 3 = (x+1)2 + 2 This is the graph resulting from shifting the
curve of the squaring function one unit to the left and two units upward. Sketch this graph!
Example 3
Graph the curve having the parametric equations: x = 2t2 , y = 2t2+ 1
txy
-31819
-289
-123
001
123
289
31819
Plot the points indicated in the table.
Join these points. What do you get?
Eliminating the parameter
we have:y = x +1 ; x Є [0,∞) why?
The curve defined by this equation is the line segment situated on the first quadrant of the straight line which intersects the axes at (0,1) and (-1 , 0 ).
1
Example 4
Graph the curve having the parametric equations: x = sint , y = 5sint + 2
txy
002
π/217
π02
3π/2-1-3
2π02
2π +π/217
2π+ π02
2π +3π/2-1-3
Notice that the same point (x,y) may be obtained by substituting different values of t.
For example the point (0,2) is obtained by both letting t = π and t = 3π
The range of x is [-1,1] and the range of y is [-3,7]
Plot the points indicated in the table.
Join these points. What do you get?
Eliminating the parameter
y = 5x +2 ; x Є [-1,1] why?
7
11
3
2
Example 5
Graph the curve having the parametric equations: x = 3cost , y = 3sint
txy
030
π/203
π-30
3π/20-3
2π30
2π +π/203
2π+ π-30
2π +3π/20-3
Plot the points indicated in the table.
Join these points. What do you get?
The range of x is [-3,3] and the range of y is [-3,3]
Eliminating the parameter
x2 + y2 = 9 why?
What does this equation represent?
X2+y2=9
Example 6
Graph the curve having the parametric equations: x = 3cost , y = 3sint; t ε [0,π]
txy
030
π/203
π-30
Plot the points indicated in the table.
Join these points. What do you get?
The range of x is [-3,3] and the range of y is [0,3]
Eliminating the parameter
x2 + y2 = 9 ; y ≥0
What does this equation represent?
X2+y2=9, 0≤y
Example 7
Graph the curve having the parametric equations: x = 2cost , y = 3sint
txy
020
π/203
π-20
Graph the curve having the parametric equations: x = 2cost , y = 3sint
txy
3π/20-3
2π20
Plot the points indicated in the table.
Join these points. What do you get?
The range of x is [-2,2] and the range of y is [-3,3]
Eliminating the parameter
X2 / 4 + y2 /9 = 1 How?
What does this equation represent?
X2 / 4 + y2 /9 = 1Is the equation for an ellipse centered at the origin and with
the vertices:
( 2 , 0) , ( - 2 , 0) , ( 0 , 3) , ( 0 , - 3)
X2 / 4 + y2 /9 = 1
Example 8
Graph the curve having the parametric equations: x = 2cost , y = 3sint; t ε [π, 2 π]
txy
π- 20
3π/20- 3
2π20
Plot the points indicated in the table.
Join these points. What do you get?
The range of x is [-2,2] and the range of y is [-3,0]
Eliminating the parameter
X2 / 4 + y2 /9 = 1 ; y ≤ 0 How?
What does this equation represent?
X2 / 4 + y2 /9 = 1It is the equation for an lower half ellipse centered at the
origin and with the vertices:
( 2 , 0) , ( - 2 , 0), ( 0 , - 3)
0;194
22
yyx
Homework
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