1027
Chapter 10
ANALYTIC GEOMETRY
Section 10.1: Parabolas
1. (a)
The relation
()
2
24
yx
−= +
has vertex
( )
4, 2
−
and opens up, so the correct
choice is D.
(b)
The relation
2
42
has vertex
( )
2, 4
−
and opens up, so the correct
choice is B.
(c)
The relation
2
−=
− +
has vertex
( )
4, 2
−
and opens down, so the correct
choice is C.
(d)
The relation
2
has vertex
( )
2, 4
−
and opens down, so the correct
choice is A.
(e)
The relation
2
xy
has vertex
( )
2, 4
−
and opens to the right, so the
correct choice is F.
(f)
The relation
2
has vertex
( )
4, 2
−
and opens to the right, so the
correct choice is H.
(g)
The relation
2
has vertex
( )
2, 4
−
and opens to the left, so the
correct choice is E.
(h)
The relation
2
has vertex
( )
4, 2
−
and opens to the left, so the
correct choice is G.
2. (a)
Since the variable
x
is squared, we know
that the parabola will either open up or
down. Since
y
is isolated and the
coefficient of the
2

x
term is positive, we
know the parabola must open up. Thus,
the correct answer is B.
(b)
Since the variable
x
is squared, we know
that the parabola will either open up or
down. Since
y
is isolated and the
coefficient of the
2

x
term is negative,
we know the parabola must open down.
Thus, the correct answer is D.
(c)
Since the variable
y
is squared, we know
that the parabola will either open to the
left or to the right. Since
x
is isolated and
the coefficient of the
2

y
term is
positive, we know the parabola must open
to the right. Thus, the correct answer is A.
(d)
Since the variable
y
is squared, we know
that the parabola will either open to the
left or to the right. Since
x
is isolated and
the coefficient of the
2

y
term is
negative, we know the parabola must
open to the left. Thus, the correct answer
is C.
3.
22
100
0
10
x y
x
y
−= ⇒ =− ⇒
=−
−
+ ⇒ − =−
−
The vertex is
( )
0,0 . The graph opens to the
left and has the same shape as
2
=
. The
domain is
( ]
,0 .
−∞
The range is
( )
,.
−∞ ∞
The
graph is symmetric about its axis, the
horizontal line
0
y
=
(the
x
axis). Use the
vertex and axis and plot a few additional
points.
x
y
4
−
2
1
−
1
0
1
−
1
−
4
−
2
−
4.
02
2
0
x
y
x
y
−= ⇒= +⇒
=− +⇒−=−
The vertex is
( )
2,0 . The graph opens to the
right and has the same shape as
2
=
. It is a
translation 2 units to the right of the graph of
2
=
. The domain is
[ )
2,
.
∞
The range is
( )
−∞ ∞
The graph is symmetric about its
axis, the horizontal line
0
y
=
(the
x
axis).
Use the vertex and axis and plot a few
additional points.
(
continued on next page
)