(b)
y
5
t
5
(
2
ˇ
t
w
)
2
5
x
2
The parametrized curve traces the left half of the
parabola defined by
y
5
x
2
(or all of the curve defined
by
x
52
ˇ
y
w
).
15. (a)
[
2
1, 5] by [
2
1, 3]
Initial point: (0, 0)
Terminal point: None
(b)
y
5
ˇ
t
w
5
ˇ
x
w
The parametrized curve traces all of the curve defined
by
y
5
ˇ
x
w
(or the upper half of the parabola defined
by
x
5
y
2
).
16. (a)
[
2
3, 9] by [
2
4, 4]
No initial or terminal point.
(b)
x
5
sec
2
t
2
1
5
tan
2
t
5
y
2
The parametrized curve traces all of the parabola
defined by
x
5
y
2
.
17. (a)
[
2
3, 3] by [
2
2, 2]
No initial or terminal point. Note that it may be
necessary to use a
t
-interval such as [
2
1.57, 1.57] or
use dot mode in order to avoid “asymptotes” showing
on the calculator screen.
(b)
x
2
2
y
2
5
sec
2
t
2
tan
2
t
5
1
The parametrized curve traces the left branch of the
hyperbola defined by
x
2
2
y
2
5
1 (or all of the curve
defined by
x
52
ˇ
y
2
w
1
w
1
w
).
18. (a)
[
2
6, 6] by [
2
5, 1]
No initial or terminal point. Note that it may be
necessary to use a
t
-interval such as [
2
1.57, 1.57] or
use dot mode in order to avoid “asymptotes” showing
on the calculator screen.
(b)
1
}
2
y
}
2
2
2
x
2
5
sec
2
t
2
tan
2
t
5
1
The parametrized curve traces the lower branch of the
hyperbola defined by
1
}
2
y
}
2
2
2
x
2
5
1 (or all of the curve
defined by
y
52
2
ˇ
x
2
w
1
w
1
w
).
19. (a)
[
2
9, 9] by [
2
6, 6]
No initial or terminal point.
(b)
y
5
4
t
2
7
5
2(2
t
2
5)
1
3
5
2
x
1
3
The parametrized curve traces all of the line defined by
y
5
2
x
1
3.
20. (a)
[
2
6, 6] by [
2
4, 4]
No initial or terminal point.
(b)
y
5
1
1
t
5
2
2
(1
2
t
)
5
2
2
x
52
x
1
2
The parametrized curve traces all of the line defined by
y
52
x
1
2.
21. (a)
[
2
3, 3] by [
2
2, 2]
Initial point: (0, 1)
Terminal point: (1, 0)
(b)
y
5
1
2
t
5
1
2
x
52
x
1
1
The Cartesian equation is
y
52
x
1
1. The portion
traced by the parametrized curve is the segment from
(0, 1) to (1, 0).
22. (a)
[
2
2, 4] by [
2
1, 3]
Initial point: (3, 0)
Terminal point: (0, 2)
(b)
y
5
2
t
5
(2
t
2
2)
1
2
52}
2
3
}
(3
2
3
t
)
1
2
52}
2
3
}
x
1
2
The Cartesian equation is
y
52}
2
3
}
x
1
2. The portion
traced by the curve is the segment from (3, 0) to (0, 2).
23. (a)
[
2
6, 6] by [
2
2, 6]
Initial point: (4, 0)
Terminal point: None
(b)
y
5
ˇ
t
w
5
4
2
(4
2
ˇ
t
w
)
5
4
2
x
52
x
1
4
The parametrized curve traces the portion of the line
defined by
y
52
x
1
4 to the left of (4, 0), that is, for
x
#
4.
Section 1.4
21