PRECALCULUS
CHAPTER 2 STUDY GUIDE
f dEd yoti learn?
n i"3
Rerriew Exercises
1-6
to analyze graphs of quadratic functions
to write quadratic functions in standard form and sketch their graphs
7-18
19-22
to use quadratic functions to model and solve real-life problems
3..2
23*28
to use transformations to sketch graphs of polynomial functions
to use the Leading Coefficient Test to determine the end behavior
graphs of polynomial functions
29-32
33-38
to use zeros of polynomial functions as sketching aids
to use the lntermediate Value Theorem to help locate zeros of
39-42
How to use long division to divide polynomials by other polynomials
43-48
Haw to use synthetic division to divide polynomials by binomials
49-56
:,flow to use the Remainder Theorem and the Factor Theorem
57-60
'llow to use polynomial division to answer questions about real-life problems
::;Sgctie$ ?"5
'E How to use the Fundamental Theorem of Algebra to determine the
f numbers of zeros of polynomial functions
irF
':'_
Ll
il
61
80-85
Uow to find rational zeros of polynomial functions
86*93
How to find conjugate pairs of complex zeros
94,95
ttow to find zeros of polynomials by factoring and using Descartes's Rules of Signs
96-1 0i
.Seefion 2.6
i,'!
How to find the domains of rational functions
.:'tr How to find the horizontal and vertical asymptotes of graphs of rational functions
'
1
02-1 05
1
06-1 09
L-l How to analyze and sketch graphs of rational functions
11A-121
n
How to sketch graphs of rational functions that have slant asymptotes
122-125
How to use rational functions to model and solve real-life problems
126*129
"fl
SELF TEST
this test as you wourd take a test in crass. After you
are done, check your work
against the answers given in the back of the book.
Take
1. Describe how the graph ofg differs from the gtaph of
f(x) : az.
(a)s(") :2-xz
(b) g(x) : (, - 1)'
2. write the equation in standard form of the parabora shown
in the figure.
3. Thepathof aballisgivenby.y: _**, * 3x * S,whereyistheheighr(in
feet) of the ball and
was thrown.
(a) Find
r
is the horizontai distance (in feet) from where the
barl
rhe maximum height of the ball.
(b) Find the disrance the ball travels.
4. Determine the right-hand and left-hand behavior of the graph
of the function
h\t) : -;t' + 2tz.Then sketch its graph.
5. Divide by long division.
6. Divide by synthetic
division.
3x3+4x-l
x"+1
2x4-5x2-3
x-2
7. Use synthetic division to show that x :
-6 ls u solution of the equation
4x3 - x2 - 12x * 3 : 0. Use the result
to factor the polynomial
completely and list all the real solutions of the equation.
H ln Exercises
9 and l',.rist
at the possibre rSti-gnar zeros of
the function. use
graphing utility to graph
the fun.tiJn unJfina aU the
rational zeros.
:2ta - k3 + l6t-24
11. Findall zeros of
f(x) : x4 * x3 *
9. sQ)
il"TJff::J"1and
13'
h(x):3x5+ 2xa _ 3x_ z
2x2 _ 4x _ g giventhatf(Zl) :0.
10.
find a polvnomial function
with inteser coefficienrs that has
12.0,3,3+i.3_i
13.
1+ Jit,t_.4i,2,2
ln Exercises 14-16,find the
domain of the function and
identify any asymptotes.
M.r:-z-_
4
x
ln Exercises 17 and
17.hG\:4 -,
x2
ls.
1g,
a
f(x):
?--2
ff
rc.
ge) _.u2
! 2x -
3
x_2
graph the function. rdentify
any intercepts and asymptotes.
18. s(*)'
: *' + 2
x-1
PRECALCULUS
CHAPTER 1 STUDY GUIDE
learn?
$,.q, wh"ri .i
ti9
Ue.tWeeofu,v.3riabl€s,arefunqtion5,
use function notation and evaluate functions
ooOAins:o.f:fun.ffi!.,'.',:' ,1:.'..: ',' '''.
usef unttig,$.tl@e!gnd9qJv9req,t-Jlfe:p.robtems
nnO the
15
At
SELF TEST
test as you wourd take a test in crass.After you are done, check your
work
againstthe answers given in the backofthe book.
Take this
ln Exercises 1-3, use intercepts and symmetry to sketch the graph
of the equation.
7.v:a-1,
2.y:4-il-l
3.y:4-(x-))z
ln Exercises 4 and 5,find an equation of the rine passing through the given
points.
4. (2, -3),(-4,e)
s. (3.0
8), (7,
-6)
6. Find an equation of the rine thar passes through the point (3, g) and
is
parallel to and (b) perpendicular to the line 4x -t- jy- : _
S.
-
7. Evaluatef(r)
I
ln Exercises
8. ,f(") :
: 'T+9
ffareach
value: (a)
/(7)
(b)
f(-5)
@t
(a)
f(x _ 9)
and 9, determine the domain of the function.
a100
-?
e.f(*):l-x+61 +2
ffi ln Exercises'ro-12,
(a) find the zeros of the function, (b) use a graphing
utirity to
graph the function, (c) approximate the intervals over which
the function is increasing, decreasing, or constant, and (d) determine whether the function
is even, odd, or
neither.
70.
f(x):
Zx6
+
5xa
- x2
13. Skerch the graph of f (x)
11.
f(x): qx_/T-
IZ.
f(x): lr + 5l
x < -3
: [t*.* ',
l4x,-7.
x>-3'
ln Exercises I 4-1 6, sketch a graph of the function.
7a. hG)
:
*x3
-7
ts.
ln Exercises 17-2$,letf(x)
value.
17.
(f +
g)(2)
ls.
gyz
=
("f
h(x)
-
: -!GT
7 and g(x)
- cX-3)
+
8
= _x2 _
le.
M. hlx): jlx + rl _ :
4x
*
(/exO)
5. Find the indicated
20.
(s./X*
ln Exercises 2'l-23,find the inverse function, if possible.
2I. f(x):
x3 +
8
22.
f(r):
lxz
- 3l + 6
23.
f(x)
:
JX-J X
8
1)
PRECALCULUS
CHAPTER 4 STUDY GUIDE
d*# y*c* $earrc?
describe angles
Keview Exercrses
1-4
5-20
use radian and degree measure
ilto use angles to model and solve real-life problems
21,22
4""r
to identify a unit circle and its relationship to real numbers
to evaluate trigonometric functions using the unit circle
23-26
to use the domain and period to evaluate sine and cosine functions
to use a calculator to evaluate trigonometric functions
31-34
27-30
3s-38
r+,ii
to evaluate trigonometric functions of acute angles
39-42
to use the fundamental trigonometric identities
to use a calculator to evaluate trigonometric functions
43-46
47-52
to use trigonometric functions to model and sotve real-life problems
53,54
tr
"!+
s5-68
ow to evaluate trigonometric functions of any angle
to use reference angles to evaluate trigonometric functions
69*74
to evaluate trigonometric functions of real numbers
75-82
eY; 4"5
to use amplitude and period to sketch the graphs of sine and
83-86
:cosine functions
,How to sketch translations of graphs of sine and cosine functions
87-90
How to use sine and cosine functions to model real-life data
91,92
**
4.6
How to sketch the graphs oftangent and cotangent functions
93-96
How to sketch the graphs of secant and cosecant functions
How to sketch the graphs of damped trigonometric functions
i,*{t
d+"7
1
How to evaluate the other inverse trigonometric functions
109-120
How to evaluate the compositions of trigonometric functions
121-128
I
03-1 08
4,$
How to solve real-life problems involving right triangles
','U How to solve real-life problems involving directional bearings
,
101,142
How to evaluate the inverse sine function
.j:lrSec?{etr
.I
97-100
How to solve real-fife problems involving harmonic motion
129,130
131
132
PRECALCULUS
CHAPTER 4 STUDY GUIDE
d*# 3r*u $eanr:?
Revieus Exercises
1-4
to describe angles
use radian and degree measure
21,22
4,€
to identify a unit circle and its relationship to real numbers
to evaluate trigonometric functions using the unit circle
23-26
to use the domain and period to evaluate sine and cosine functions
to use a calculator to evaluate trigonomeftic functions
31-34
3s-38
t,
27-30
."5
to evaluate trigonometric functions of acute angles
to use the fundamental trigonometric identities
39-42
to use a calculator to evaluate trigonometric functions
47-52
to use trigonometric functions to model and solve real-life problems
53,54
:+-
c*
5-20
to use angles to model and solve real-life problems
43-46
*
to evaluate trigonometric functions of any angle
55-68
to use reference angles to evaluate trigonometric functions
69-74
to evaluate trigonometric functions of real numbers
75-82
d*"5
to use amplitude and period to sketch the graphs of sine and
.:Eosine
functions
83.-86
,,How to sketch translations of graphs of sine and cosine functions
87-90
How to use sine and cosine functions to model real-life data
91,92
*lc 4-6
93-96
How to sketch the graphs oftangent and cotangent functions
How to sketch the graphs of secant and cosecant functions
How to sketch the graphs of damped trigonometric functions
97-100
101,102
4.7
How to evaluate the inverse sine function
1
How to evaluate the other inverse trigonometric functions
109-120
;:,:,[ How
to evaluate the compositions of trigonometric functions
03-1 08
121-128
Sel"ton +"g
,,
I
How to solve real-life problems involving right triangles
E
How to solve real-life problems involving directional bearings
131
How to solve real-life problems involving harmonic motion
132
' tr
129,130
SELF TEST
Take this test as you would take a test in class. After you are done, check your work
against the answers given in the back ofthe book.
1.
5rf 4 radians.
(a) Sketch the angle in standard position.
(b) Determine two coterminal angles (one positive and one negative).
(c) Convert the angle to degree measure.
2. A truck is moving at arate of 90 kilometers per hour, and the diameter of its
Consider the angle of magnitude
wheels is
1 meter. Find the angular speed of the wheels in radians per minute.
the exact values of the six trigonometric functions of the angle 0 shown
in the figure.
3. Find
4. Given that tan e : ), nna the other five trigonometric functions of g.
5. Determine the reference angle 0'of the angle 0 : 290'and sketch 0
in standard position.
and 0,
6. Determine the quadrant in which 0lies if sec g < 0 and tan 0 > O.
7. Findtwovaluesof gindegrees (0 < g < 360")if cos d : *J1/2.(Donot
use a calculator.)
8. use a calculator to approximate two values of g in radians (o < e < 2r) if
csc d
:
i.030. Round the result to two decimal places.
ln Exercises 9 and 10, find the remaining five trigonometric functions of 0 satisfying
the conditions.
9.cosg:!,tang<0
ln Exercises
1
1
10. sec
0: -+. sin d > 0
and 12, graph the function through two full periods without the aid
of a graphing utility.
11.s(x)
/^\
:-2sin{*-+}
\ 4/
12.
f(a)
:
)tunzo
In Exercises 13 and 14, use a graphing utility to graph the function. lf the function is
periodic, find its period.
13. ),
:
sin2nx
15. Find a, b,
/
*
2
cos
nx
and c for the
matches the figure.
14.
function/(r)
16. Find the exact value of tan(arccos
l)
y:6e-012t cos(o.zsr), o < t<32
:
a sin(bx + c) such rhar rhe graph of
witrrout rhe aid of a calcularor.
17. Graph the function/(r) : Zarc.in(jr).
18. A plane is 80 miles south and 95 miles east of an airport. what bearing
should be taken to fly directly to the airport?
19. write the equation for the simple harmonic motion of a ball on a spring that
starts at its lowest point of 6 inches below equilibrium, bounces to its maximum height of 6 inches above equilibrium, and returns to its lowest point in
a total of 2 seconds.
PRECALCULUS
CHAPTER 5 STUDY GUIDE
Wtzat did you learn?
5,1
n How to recognize and write the fundamental trigonometric identities
tr How to use the fundamental trigonometric identities to evaluate
Scction
Reuiew Exercises
1-6
7-23
trigonometric tunctions, simplify trigonometric expressions, and rewrite
trigonometric expressions
F-
F
A
-!--5"c
5ecElon
I
x
How to plan a strategy for verifying trigonometric
How to verify trigonometric
identities
identities
24-31
24-31
>ecnon 5,5
I How to use standard algebraic techniques to solve trigonometric
32-37
equations
n
I
n
type
angles
functions to solve trigonometric
How to solve trigonometric equations of quadratic
38-41
How to solve trigonometric equations involving multiple
42-45
How to use inverse trigonometric
equations
46-49
bectron 5.4
I How to use sum and difference formulas to evaluate
trigonometric
50-63
functions
I
How to use sum and difference formulas to verify identities and
solve
trigonometric equations
SELF TEST
Take this test as you would take a test in class. After you are done, check your work
against the answers given in the back of the book.
I : ] and cos 0 < 0, use the fundamental identities to evaluate the
other five trigonometric functions of 6.
1. If tan
2.
Use the fundamental identities to simplify csc2 B(1
3. Factor and simplify
4. Add and simplify
'
-{
sec4
-
tana
-
cos2 B).
x
sec'x + tan'x
. 99.
TY
sin 0 cos 0'
5. Determine the values of 0, 0 < 0 < 2r, for which tan 0 :
is true.
- JT;A 0 - |
H 6. Use a graphing utiiity to graph the functions y1 : cosx * sin xlanx and
)2 : sec x. Make a conjecture about y, and yr. Verify the result analytically.
ln Exercises 7-12,verify the identity.
7. sin9sec0:tan9
csca*secd
a * cos a
11. sin(nrr + g) : (12. (sin .r * cos x)z :
-:cota
^
9.
i tana
sin
8. sec2xtanzx *sec2;r:sec4x
( n\
10.
-"' cos{x+
-""\" :l : -sinx
1),' sin 0, n is an integer.
1
t
sin 2x
Zl
64-67