3.5 Parametric Equations Definition: ,
, where f and g are continuous on an Let C be the curve consisting of all ordered pairs interval I. The equations ,
for t in I, are parametric equations for C with parameter t. A curve defined parametrically has an orientation which is the direction determined by increasing values of the parameter. As the parameter t increases, the point , traces the curve and the direction the point takes is the curves orientation and is described by placing arrowheads on C. 1. Make a table of values and draw the graph of the curve C defined by the parametric equations 1,
1
2 2 ,
Indicate the curves orientation. 1
2
x 2 1
3
y 0 
4
t
1 
1
2
0
1
3
1 
4
0
1
2
0
3
2
3
5
4
2
4 3
Sometimes we can eliminate the parameter between two parametric equations to find how x and y are related. This is called the rectangular equation of the curve and the process is called eliminating the parameter. In examples 2‐6, sketch the graph of the curve by eliminating the parameter and determining its equivalent rectangular equation. Indicate the curves orientation. 2. 1 2 ,
1
, 1
4 Solution: 1 2 also 1
4 4 2
1
1
1
1
7 3 3
7
x
7
3
y
5 0
1,
√
Solution; Since √ ,
0 and 3. 0 √
1 , we have 1 1 1 1 √
1 x
1
y
0
2
3
3
8
4. 2
,
1 2 Solution: 2
2
1 2 1 2 2
1 4 2 5 2 Note since 2
we have 2 5. 2 cos ,
3 sin , 0
2 Solution: To eliminate the parameter we make use of the Pythagorean Identity 2 cos , 3 sin cos
sin
1 1 2
3
1 1 This is an equation of an ellipse center (0,0) 6. sec ,
tan ,
Solution: To eliminate the parameter we make use of the Pythagorean Identity 1
1
1 This is an equation of a hyperbola center (0,0) Note: since sec
we have 1
1 ,
If C is a curve given parametrically by ,
tangent line to C at a point , then the slope of the is 7. Let C be the curve given by the parametric equations equation of the tangent line to the graph of the curve at Solution: , ,
3 ,
1
9
5
1. Determine the 5
1
8
6,4
7
, 7
3 ,
2. 10
1
2, 7 2 2
9
1
9
61
9
Second Derivative in Parametric Form: =
In example 8 & 9, find an equation for the line tangent to the curve at the point defined by given value of t. Also, find the value of 4
8. 5,
2
at this point. 3, @
1 , ,
4
5,2
3
4
, 8
1
32
3
5
, 5
5,2
1 1
4
8
1
4
1
32
1
32
1,5 9. 2
3,
, &
1 , Solution: ,
1 3,
2
5,1 , 1
1
5 4 8
1
32
1
4
8
1
4
1
32
1
32
The parametric equations for the line segment with endpoints ,
,
, 0
and ,
1 are 10. Find the parametric equations for the line segment with endpoints 2,5
3, 4 Solution: ,
, 0
1 2
3
2 ,
5
4 5 , 0
1 2 5 ,
5 9 , 0
1 11. Find the parametric equations for the half line with initial point 3,4 and passing through the point 1,3 . ,
Solution: 3
1
3 ,
4
3 4 ,
0 3 4 ,
4
,
0