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Conics and other solutions of equations in two variables
18.) Give the equation for a line of slope 2 that passes through the origin.
7
/
19.) Give the equation for a line of slope 2 that passes through
the point (3,4).
(-2(x-3)
7
20.) Give the equation of the unit circle.
‘S
21.) Give the equation of the circle of radius 4 centered at the origin.
yo’
-LI.
(QV
j-L ‘s
s
wte. 1
6.) Give the equation øf the circle of radius 4 centered at the
Give
6.)
the equation øf the circle of radius 4 centered at the origin.
7.) Give the equationthe circle of radius
4 centered
at the of
point
(5,7). of radius 4 centered at the poi
22.) Give
the equation
the circle
7.) Give the equationthe circle of radius 4 centered at the point (5,7).
7
22.) Give the equation of the circle of radius
4 centered at the point (5, 7).
7
6.) Give the equation øf the circle of radius 4 centered at the origin.
6.) Give IIthe equation øf the circle of radius
4 3centered at the origin.
22/a
3
;/
2
(x5)(y-7)
4 circle of radius 4 centered at the
7.) Give the equationthe
22.) Give the equation o
4
7.) Give the equationthe circle of radius 4 centered at the point (5,7
8.) Give the equation,the ellipse obtained by starting with the unit circle,
23.) Give
thetheequation
of the ellipse obtained by starting with th
Q
3
8.) Give the equation,the
ellipse
obtained
starting
with
and then
scaling
the x-axis
by by
3, and
the y-axis
byunit
2. circle,
and then scaling the x-axis by 3, and the y-axiscircle,
by 2. and then scaling the x-axis by 3, and the y-axis by 2.
-
7
I
4
7
23.) Give the equation of the ellipse obtained by starting with the unit
3
7.) Give
equationthe
circleby
of 3,radius
4 centered
at 2.the of
point
(5,7). of radius
circle,
andthe
then
scaling the x-axis
the
y-axis
3 circle
andGive
by
22.)
the equation
the
7.) Give the equationthe circle of radius 4 centered at the point (5,7).
5
‘3
4
(x)Z
a8.) Give
‘3 the equation,the
ellipse obtained by starting with th
7
23.) Give
thetheequation
o
8.) Give the equation,the
ellipse
obtained
starting
with
and then scaling
the x-axis
by by
3, and
the y-axis
byunit
2. circ
7
and then scaling the x-axis by 3, and the y-axiscircle,
by 2. and then scali
2
o Carl
oc from
2
24.) Give the equation of the3 ellipse
#8, shifted right by 4 a
3
9.) Give the equation the ellipse
from
right
by
+.iS
shifted
4
and
AJ(;t€.
up by 5. #8,
up by 5.
4
2
2 * IL
X
7
-4--—
4
8.) Give the equation,the ellipse obtained by starting with the unit circle,
‘3
24.)the
Give
the equationellipse
of the obtained
ellipse
from starting
right
by 4 and
23.)
Give
the
equation
of the ellipse obtained
5
#8, shifted
8.) Give
equation,the
with
the
and then scaling
the x-axis by by
3, and the y-axis
byunit
2. circle,
‘3
upby5.
and then scaling the x-axis by 3, a
and then
scaling the x-axis byMatch
3, 3and
y-axiscircle,
by 2.their
thethe
functions
with
graphs.
(i):
N.
/
7
5
3
0
0
—
)
33.) sin(x)
34.) sin (z + 2
35.) sin(2x)
2
24.)
Give
the
36.)9 sin
37.) sin(x) + 1
sin(x) o
38.) equation
9.) Give the equation the
ellipse
from
#8, shifted
up by 5.
39.) cos(x)
sin
(x
40.)
41.)
2sin(x)
up by 5.
42.) Draw
sin(x)
43.) ofsin(x)
7 xy = 44.)
25.)
the set of solutions
the equation
1. sin(—x)
()
—
5
rt
—
—
0
‘3
‘3
26.) Draw the set of solutions of the equation5 x2 y 2 = 1.
A.)
B)
C.)
10.) Draw the set of solutions of the equation xy = 1.
Match
2 functions
3 the
27.) Draw2 the
x2 = 1. with
7
2 set of solutions of the equation y
25.) 11.)
Draw
thethe
setsetofofsolutions
the equation
equation
==
1.the
24.)
Give
thesin(x)
ellipse
from #8
Draw
solutions 2of the
1. equation of33.)
2 xy
x
2
28.)
Draw
the
set
of
solutions
of
the
equation
y
=
x
.
9.) Give the equation the ellipse
from
right
by
shifted
4
and
#8,
up by 5.
36.) sin
26.) 12.)
Draw
thethe
solutions
equationy
Draw
setofof
setup
of the
the equation
2 = 1.= 1.
solutions
by
2x
5.
39.) cos(x)2
29.) Draw the set
2! 7of solutions of the3 equation x = y .
2.
42.) Draw
sin(x)
2
25.)
the set of sol
Draw
solutions of the
equation
.
2
y =y
27.) 13.)
Draw
thethesetsetofofsolutions
the
equation
= 1.
2x
5
26.) Draw the set of sol
Draw
solutions of
.x
2
y
-2
28.) 14.)
Draw
thethesetsetofofsolutions
of the
the equation
equationx =y =
.
2
A.) their graphs.
B
Match
the
functions
with
3
10.)
Draw
the
set
of
the
equation
solutions
xy
of
= 1
D.)
E.)
F.)2 the set of sol
27.) Draw
29.) Draw the set of solutions of the equation
x
.
2
y
=
33.)plane
sin(x)
34.) sin (z +
15.) R/
4=
is the rotation of11.)
angle
the
by set
Draw the
of solutions of the equation
2
x
//J
28.)
the+set
36.) sin
sin(x)
37.)Draw
1 of sol
If H C R
2 is the set of solutions of xy =
then
draw
R_(H).
cos(x)
(x
12.)1,39.)
Draw
the set
of solutions 40.)
of thesin
equation
2
y
29.)
Draw
the
set
of sol
2!
42.) Draw
sin(x)
43.) ofsin(x)
25.)
the
set
of
solutions
the
equation
2.
2
16.) Let P the set of solutions of y = x
2 from
equation of the equation y = x
13.) #13.
DrawW1±att
the set the
of solutions
.
2
—
()
—
—
(
0
()
)
—.
—
—
—
30.) R_/
4
is the rotation of the plane by angle
=
If H C R
2 is the set of solutions of xy
=
—.
1, then draw R_
(H).
1
31.) Let P be the set of solutions of y = x
2 from #28. What’s the equation
for P shifted right by 2 and up by 3?
7
J
\
—i
2
(-3)L-2’)
1)
2
Trigonometry
32.) What is the distance between the points (2, 5) and (3, 8)?
+9
33.) Find the length of the unlabeled side of the triangle below.
7
So
jj’
8
34.) Find sin(s), cos(), and tan(O) for the angle
S
3
(a):
S
coqe)
1
I,
ao.s
7 5c c>O.
J7i
given below. (3 points.)
Match the functions with their graphs.
35.)
38.)
41.)
44.)
sin(x) A
sin
3
cos(x)
sin(x)H
36.)
39.)
42.)
45.)
()
A.)
I
)
sin (x +
sin(x) + 1
sin (x
—sin(x)
37.) sin(2x)
40.) sin(x) 1
43.) 2sin(x)C_
46.)
—
0
sin(—x)E
—
B)
C.)
2
a
-‘?r
n_
-
2
—I
—I
-2
-z
D.)
E.)
-,r\J/
*1
a
—2
F.)
2
Z
2.
iT
2.
-z
-2
-z
G.)
H.)
I.)
2
2
2
ñ
‘1
\Jr
-
—t
-2
J.)
2
a
.—t
2
71f\
-2-
Match the functions with their graphs.
47.) f(x)
=
fsin(x)
lcos(x)
if x < 0;
ifx0.
A.)
48.)
()
fcos(x)
sin(x)
if x < 0;
ifx>0.
B.)
/‘C\
Co5(:
Find the values:
49.) arccos
T
50.) arcsin(—1).. .51.) arctan(—1)—.!
52.) What’s the Pythagorean
For #53-61, graph cos(x), sin(x), tan(x), sec(x), csc(x), cot(x), arccos(x),
arcsin(x), and arctan(x).
************************************************
Complex numbers
62.) Find (2+i3)+(4+i5).
63.) Find (2 + i3)(1 + i4).
(j(3,5’)
2(i) ÷ Z(z•t) z’3(i
)+
Z-i’8+i3—i
IO-fj/I
i8
64.) What’s the norm of 7 + i3.
)7-i3)
7
2 +3L+
65.) Write 5 + i2 in polar coordinates.
I+iz1
\
66.) Find 2(cos(3) + i sin(3))4(cos(8) + isin(8)).
(cos (3)+Is(+)
Ca5(i1)+ith (ii
67.) Find (+i$)
4
(q
—
—
a1
68.) Find
(
I
(— + i sin ( —
Draw an X on the number 3(cos () + i sin ())z.
69.) Draw a dot on the number 2(cos
70.)
(The number z is drawn on the answer sheet.)
************************************************
Equations in one variable
For #71-76, give the real number solutions of the equations. If there is no
solution, explain why there is no solution. For at least one of the problems,
the domain of the equation will play an important role. Worth 2 points each.
71.) ()2+/_6O
j
C°
c.z’)
/
51 r ce. we
Ca
sra?c
-
rool
a1V bjil,er.
3
0. yt
aL,
Rc,a
4’:
‘4((
Tkt-
-
ac
IS,
kas
-
S
-Th2. &-f
I ton
aci 1tw.s, x:L1.
72.) loge(x 1) + loge(x + 3)
—
DowLcn’ x-(
I
-
C—
-I±iJ5’ -1i5
-
ZCa
13•I,
’J
4
Q
=
.S ()
,
2
0
x.+S’O.
-
x>I
2
o....j
aL, 6z, c—’t
73.) e
21
=
—3
/
x—i-J?
SoLu.kov
.
s
ic
-L-Ji.____
‘0
otwtv
z’-Sx>1...
‘C;
f)(x43) a
Xt+2Lt
acI
S )Dc.YSih\’L.
7
aft alw’
z
2
a
74.) (x
—
3)2
=
4
R.
o(
le
2
e
X
3
_
75.) 7
=
1
-c•
LEfT
76.) (x+1)(2x—3)= (x+1)(x—4)
V
I
Zz-3
I’
I,
I,
o
p
cz
cc
iJJ
4
t\)
o
I)
N
4
0
-
C
N
t)
0
c
I
I’
C
N)
R
0
I
I
Oa
S
I
N
I.
CD
z
Cl)
CD
Cl)
Tj
26.)
30.)
4
7
27.)
J
31.)
(3 (-2
32.)
33.)
M
---
34.) sin(s)
=
cos(6)
=
tan(9)
=
28.)
]
29.)
S
3
35.)
A
36.)
Q.
37.)
I
38.)
39.)
40.)
-
3
V
41.)
47.)
A
[3
42.)
E
48.)
43.)
C
49.)
ir,
H
EE
44.)
45.)
50.)
‘2
_1
51.)
46.)
52.)
53.) cos(9)
—
I.
2
cos(e÷ Ivi(’)
54.) sin(6)
55.) tan(s)
-2
56.) sec(s)
N,
57.) csc(6)
58.) cot(O)
2
2
n.
2.
-2
1
Ni
59.) arcsin()
60.) arccos(8)
61.) arctan(6)
--
2
I
—I
3.
—7r
62.)
66.)
63.)
—)Q+jj/
67.)
64.)
r&
68.)
65.)
(k+i)
69.) and 70.)
4:
8 (cos(ifl+js (ii)
2
22
-i7