Quick study academic pre calculus 600dpi

FUNCTIONS
A. A function is a rel ation in wh ich each el ement of the d oma in (x value
- in dependen t va ri a bl e) is pa ired w ith only one e leme nt of the ra nge
(y value ­ d epe n de nt va riable).
B. A re lation can be tested to see if it is a funct ion by the vertical lin e test.
Draw a vertical line throug h a ny g raph, and if it hi t an x- value more than
on ce, it is not a functi on . (1-4)

b. b 2 ­ 4ac = 0, exactly one rea l r ot
c. b 2 - 4ac < 0, no real roots (two disti nct imaginary roots)
I) Example l:f(x) =x2_4x+ I use

f( x)=

[!]E8rnrn

FE
A function

Not a function

A function

A fu nction

C. Linear funct io ns ta ke the form: f(x) = mx+b, or y =

m x+b where m = the. lope, and b = the y-intercept.

Examp le: f(x ) = 4 x-l , the s lo pe i 4/1 (ri e ove r run), and

th e y-intercept is -I.

D. he dis tance between two points o n a lin can be found

j( X 2 -

u ing the distance form ul a, d =

XI ) 2 + ( Y2 - YI )2

X I)

•

Vd ).

(Y 2 +
2 '

F. The s tan dard form ofa linear functi o n is 0 =Ax + By +
C. The lope is m = -A/B, and the y-intercept is -C/B.
G. The zeros of a fun ct io n a re fo und by setting y to O. and
solvi ng for x.
I. E xam ple .1: f(x) = 4x-1 (5)
2. Exam p le 2: f(x) = 6, thi s func tio n has no zero, and is

a h ri z ntal lin e th rough +6 o n the y-axi . (6)

3, Exam ple 3: x = 4, th i i not a function, because the re

is a verti ca l line through +4 on the x-axi , g ivi ng a n

in fi n ite set of va lu es for y. (7)


X;~~5

ffi

the

discriminan t is > 0, the re are two rea l roots. (15)
2) Example 2: f(x) = 2x2 + 2x + I us ing b 2 - 4 ac =
-4, since the d isc rimin ant is < 0, there are two
imaginary root . (16)
3) Example 3: f(x) = x2 + 2x + I us ing b 2 - 4ac = 0,
· ..
. 0 th
'
I
17
ere IS one rea I' \. ( )
since t1le d ISCnl11l11ant IS =
. .



g(x)

J . RatIOnal fun ction s take the fo ri: f(x ) = b (x) .

I . The pare nt fu nction i f(x) = X ·

~2
x-

+ 3 (21)

E
3
E
/

x= 1.2765957

y=-2. 106855

Two real solutions


E
4
E

a . Find th e sum : (f + g)(x), x + 2 + ~
( x + 2 ) (x - 4) + x
X2 _ X _ 8
x­
x- 4
= x - 4 ' and x f. 4.
b. Find the di fference: (f - g)(x), x + 2 - ~

(x + 2) (x - 4 ) - x x _ 3 x _ 8
x­
x- 4
=
x_ 4
• an d x f. 4.
6. Example 2: Gi ven f(x) = x + 2, g(x) = x:. 4
a. Find the product: (fxg)(x), (x + 2)( x :. 4 ) =
x 2 + 2x
---x=-;t.
an d x f. 4.
b. Find the quot ient: (
X - 4)

Example 2: f(x) = 2xJ + x2 - 2x + 3, thi function has

one rea l zero at x = -1.17, and two non -rea l roots. (8)


(x+2) (.-x- =

t

)(X),

x;x+_2 4 =

x2 -2x - S

x

x~930B511

, andx f. O.

:. 4

+ 2.

2(
4)
x ( x - 4)

+2

Example: Gi en f(x)=x+2, g(x)=
Find IfogJ(x): f ( x :. 4
( _ x_ + 2 )
x - 4

+2

=

+ 2) =
x

+

5x - 16

, and x f. 4.
x- 4
M .lnve r se function s : If og l(x) = Igo fj( x)

- b ± jb 2 - 4ac
x=-1.010638
2a
can y=2.9737903

Example: Given f(x)=2x - 4, g(x) = · +4
'-2X+4)_- 2 (X-2+4) - 4 -_x,
Ifog l(x) -_f (

x-- 0265958

y~.OlO0806

A rationalfunction

with asymptotes
at the x & y axes.

F
O E

~,

be u ed to f ind the roots of a ll quadratic eq uations.
6 . The a lue under th sq uare root symbo l is ca ll ed the

d iscrim in an t. It te ll s u the type of roots o f a quadratic

equat ion .


( 2X
and IgO fj( x) =

di st inct rea l roots
1

-

4)
2

x=-.5053192
y=.47379034
Two imaginal)!

y=.00478157

L. c om P.osition of functions:lfog l(x) = f(g(x» E
8
!J

Example 3 : f( x) = x 2 + I. this fUllction has'two nonreal roots, (9)
x=. 10638298
l. Qua dra tic funct ions take the form : f(x) = ax 2 + bx + c. y=1.0113173

I. 111e graph ofa quadrati function i called a parabola. (10)
o E
2 . ome parabolas ar e qu adrat ic eq uation , but not t
q uadratic fu nctions. (II)
3. Quadrati c functions or equat ions can have one real

so luti on. two rea l solutions, or no real so luti on. (12-14)

4 . Th vertex of a parabo la i ca ll ed its critical p oi nt.

EE

ra

ra

x=1.3829787

y=-2.671371

Two real roots,

(.26,0) & (3.73,0)


5. Example I: Given f(x) = x + 2, g ( x) == x:. 4

Example l: f( x) = 2x4 + x 2 + X + 10, has a degree of
4, there are fo ur roots (so lutions) to th is po ly nomia l. 9

t lVO

y=-.0100807
One real solullOn

. (f)

EE
tE

a. b 2 - 4ac > 0,

x = -(-4)

f< - 4 2 _ 4)/2 = 3 .732, and
- f< - 4 - 4) /2 = .267. s ince

K. Oper ations of fu nctions:
I. Sum: (f + g)(x) = f(x) + g(x)
2. Diffe rence: (f - g)(x) = f(x) - g(x)
3. P rod uct: (f x g)(x) = f(x) x g(x)
f( x)
4. QuotIent: g (x) = g (x )' g( x) f. 0

No zeros

+ cx n-2 ••• + d x + e
I . When the hi g he t power o f the func ti on is a n odd

intege r, there is at least one real zero.

2 . When the h ighe t powe r is a n even integer, there may

be no real zeros .

3. Both type can have imag inary roots of the form a + bi.
x=-1.170213
4. T h hi ghest p wer of a po lyno mi a l w ith o ne vari able is
y=.0100806
ca ll ed it degr ee.

=

x = -(-4) +

6 . Exa mple 3: f(x) =

H. Polyno mial function s take the form : f(x) = ax n + bx n- I

). The quadratic equat ion f(x)

2a

2. Th e g raph of these functi ons cons ist of two pan s, one
in quadrant I, and one in quad rant 3.
3. The bran hes of rational functi on approac h line
call ed asy mptotes. (18)

x

x=-.9574468
4. Exa mple I: f(x) = x + 3 (19)

y=7.9737903
No real solutions
3

5. Exam ple 2: f(x) = x (20)

E. The mid-point of a line segment can be found us ing the
..
( X2 +
m id -po lllt for m ula,
2

-b ± /b 2 _ 4ac

+4

= x.

x=.02659573
y=-.0100806
The asymptotes
are the axes

y1=x~2x+1

One real rOO! (-.93,0)

N. Families of functions : Graphs of fu nction fami lies. Changes in va lues of
the parent affect the appearance of the paren t g raph. A par ent g ra p h is
the basic grap h in a fam ily. II the other fa mily m mbers move up, down,
left. right, o r turn based on changes in va lues .
I. Polynomia l fun ctions 1:
3. A bsolute value fu nctions:
a. f(x) = Ixl (38)
a. f(x) = x 2 (22)
b. f( x) = 2x2 (23)
b. f(x) = -I xl (39)
c. f(x) = .5x2 (24)

c. f{x) = 12xl (40)

_
2

d.( f(x) = 1.5xl (41)

d.f(x) - -x (25)
e .f(x) = Ix + 21 (42)

e . f(x) = x 2 + 2 (26)

f. rex) = Ix - 21 (43)
f f(x) = x 2 ­ 2 (27)
g . f(x) = Ixl + 2 (44)
g. r(x) = (x + 2)2 (28)
h. f(x) = Ixl- 2 (45)

h.r(x) = (x - 2)2 (29)
2. I>olynom ial function s 2:
a. f(x) = x3 (30)
b. f(x) = _x3 (31)
c. f{ x) = x3 + 2 (32)

d.f(x) = x3 - 2 (33)

e. f( x) = 2x3 (34)
f. f( x) = .5x3 (35)
g. f( x) = (x + 2)3 (36)
h.f(x) = (x - 2)3 (37)

A. Rectangula r coordinates arc o f the form (x,y), and arc pl otted on the
Cartes ian coord in ate syste m.
B. Poin ts are p lotted w ith two va lues, one the absci sa and the other th

ordinate.
C. T he absc issa i the x-va lue, call ed the domain. and th e ord inate is the

E
S f]

y- va lue, called the range.

/P

D. Many di ffe rent shape and func tion can be drawn on the
Ca rt e ian system.
E. Here is a g ive n ang le, orig inating fr

the

III

-axi

and

rotating counter-clockwise. Th is ang le is re prese nted by a

P(4.6)

li ne segment ori ginati ng at t.he o ri gin, and extend in g to a
given poin t (P). (46)
R Pola r coo rdinates are o f the fo rm P( r, 9), where r

E
7 B
,p

= the

I

(.

radiu , the di stance from the o ri g in (0,0) to I) (a g iven
po int), and El = the magnitude o f an ang le.
I. If r is pos iti ve,

e is

P(H,O)

the meas ure o f any a ng le in

sta nd ard pos ition th at has segm ent 0 ,1' a its te rmina l
s ide.

2. If r is negative, El is the measu re of any ang le that has
ray o ppos ite segmen t O, P as its ter mina l side.

th

(47&48)

22

EEJ

P(-r,9)

G. Graphing w ith polar coordi nates :

I. Exa mp le I: 1'(4, 120 d egr ees) (49)

2. E xa mp le 2: P ( 4. ~ ) (50)
H. O ne angle graphed w ith polar coordinates can

I'

presen t

seve ra l angle .
I. If Pis a po int w ith polar coo rdinate (r, 9), then l' can
also be g raphed by t he po lar coordi nates (-r,

ffi

P(4, 120)

e + ( 2x + 1)1t) or

(r, e + 2x1t), where x i any in teger.
2. E xa mlJle: S how fo ur differ nt pa ir ofp la r coo rd inate th at ca n be re prese nte d b the po int 1'(3, 60

r
a n
.

I.

degrees).
3. (-r,

e + (2 x + 1)1 80 d egrees) ~ (- (3), 60 + (1)180)

(- (3), 60 - (1)1 80),
4. (r,

= 1'(-3, 240) or 1'(-3,120)

P(4'3)

e + 360x) ~ P(3, 60 + (1)360) or 1'(3, 60 + (2(360)

= P(3, 780)

hanging from rectangula r to pola r coordina tes : The fo ll owing
fo rmulas are used to make thi s change:

j( x

f,

+ y2) , e = A rctan x > O.
2. 9 = A rc ta n
+ 11:, X < 0, a nd fj = r adian s.
3. E xample I: Find the polar coordinates f, r 1'(-2,4). r =
I. r =

=

2

f

j20 = 4.47. e = Arcta n

j

_ 2)2 + ( 4 )2

_42 + 1t = 2.03, P(4.47, 2.03).

4 . Exam ple 2: Find the po lar co rd inates for P(3 ,5). r =
5
34 = 5.83, El = A rcta n "3
= 1.03. P(5.83, 1.03).

~_~

j()2+ 5 2) =

hanging fro m polar to r ecta ng ular coordinates: The formulas used to
make thi s change are :
I. x = r cos El

e
3. ~~a~Ple 1:
2. v

=r

sin

P ( 4.

~),

x = 4 cos

( ~)

= 2,

an d 4 s in

(~)

=

3.46 - P( 2, 3.4(h.

4. Exa mple 2: P(5, 60°), x = 5 cos (60)

= -4.76. Y = 5 si n (60°) =

-1.52 =

P( -4. 76, -1.52).
K. Grapbing imaginary numbers with po lar coord inate: The polar
form of a complex number i x + yi = r(cos El + i s in e).
Example: Graph the complex number -4, + 2i, and change to polar form.
r =

;;r+yL =

j(_ 4

2

+ 2i 2)

= /16

+4

=

,fiO = 4.47, El =

rctan

( _24 ) + IT = 2.68,1)0Ia r fo rm = 1'( -4, 2 i) = 4.47(cos 2.68 + i sin 2.68)

2


A. The notation P( n,n) = the num ber o f permutations of n objects taken all
at one time.
B. The notat ion P(n,r) represents the number of perm utation of n obj cts

A. Exponential properties:
I. Multiplication: x· xh = x. + b

E xample: x 2x 4 = x6

take n r at a time P (n,r)

2. Division : (;: ) = x·- b
Example:

4. Distribu tion with d ivisio n: (

r

y

y/ = ( ;: )

= x:

18·17·16·15·14·13·12·11·10·9·8·7·6·5· 4 ·3·2·1
.
h
6.5.4.3.2.1
, not!cetatyou

y

can ca ncel 6!, leaving 18 ~ 7 = 8.89 X 10 12 choices.
Example 3: A combination lock has four tumbl ers, and i num bered I ­

5. Power of a power: (X") b = x· b

Exam ple: (x 2)3 = x6

6. Inverse power: x-I = {

x"" = ~
x

7. Root power: xI I. =
Example : XII2

20 on the dial. How many combinations are possible . P(20,4)

.j;


= j;


R


bR


B. Logarithmic Properties and Logarithmic Fo rm:

EE
FE
EB
EfJ
t§

1. Logarithmic Form: log.x = y, thi s is read a "the

exponent of a to ge t the result x is y."


1

Exa mple: log, 100 = 10, the exponent of x to get the

res ult .100 = 10 or x lO = 100.

2. Loga rithmic I)roperties :

a. M ultiplication: log.xy = log.x + log.y
b. Division: log.

y= logax - log.y

* log.x
= logaY' then

c. Power p rope rty: log.x b = b
d. rden tity property : If log.x

x= y

3. Change of Base p rope rty: I Lx, y and z are + nu mbers.

log ) z

and x and }' are n t = I , then, log, z = -I - - .
og)x
C. Solving logarithmic equations:

Exa mple I: Write log 1000 = 3 in exponential form:
10" = 1000

Example 2:

olve, log,

Ii

=

i~

x l/4

=Ii

24 - 3x. x = 6.
Example 4: log (2x + 8) - log (x + 2) = I

+8

~

~

8x = -12

(2x + 8)
10 1 = (x + 2)

~

Exam ple 5: log, 5 =

-*

~

log

IOx +20= 2x E
6 EJ

x = - L5.

~ x- 1/3 = 5 ~ x = - 1~5.

Exa mple 3: y = 2 2. - 1 - I (53)

Exam ple 4: y =

+ I (54)

Example 5: y = logz

E xample 6 : y = log2(x - 2) (56)
E xample 7: y = log2(x - 2) + 2 (57)

Example 2: How many word pattern can be form ed from Mi issippi ?
ll!
2!2!2!4! = 207,900.
Example 3: There are 24 ch ild ren who are going to play dodge ball. 10
of the m w ill start out in the circle. Ilow many ways cou ld the
remai ning children fo rm ci rcular combinations? (n-I)! = 13! =
6,227,020,800.
E. Combinations: Di ffer from permutations in that order i not a cons ider­

II! )
.
r !r!
Example 1: A book c lu b ha selected e ight books to read, how many four

ation. The number of n objects taken r at a time is C(n,r) = (

11-

group com binations are pos ible?

~

qn,r) = (

8! )
= 70
8 - 4 !4!

Exa mp le 2: How many five card hands an be dea lt from a regu lar deck
52'
o f card s? qn,r) ~ = (52 _
!5! = 2598960

5)

A. Synthetic division is a method used to make long d ivision of p Iynomial
les cumbersome .
I. It is ma inly used when you have a very long numeral r.
2. It ca n only be lIsed wi th a di i or in the form (x - n).
3. You can convert the form (3 x + n) to (x + 1/3n) and then use sy nthetic
divis ion.
B. Synthetic Division of the fo rm x2 - 3x - 54.;- x - 9.
I. Example I: T he standard way of d iv iding polynomials
3x - 54
x + 6


Btl
ttJ

Exam ple I: y = 3 ', and (1/3)' on the same grap h. (51)

Example 2: y = 92 +1 (52)


Repetitions: The number of n objects of which x & yare alike = ~.
.y.

C ircular Pattern s = (n-I)!
Example 1: How many word arrangemen ts can be made from the word
5!
radar? n = 5, x&y = a&r = 2. 2!2! = 30.

x - 9 )X 2 -

D. Graphi ng Exponent ial and Loga rithmic Equa tions:

22>-1

;gi

~


(xI /4)4 = (1i)4 ~ x = 4.
5

Example 3: logi2x - 6) = log4(24 - 3x). 2x - 6 =


2 + 8)
(x +2 ) = I

~

_ 20·19·18·17·1 6 ·15·1 4 ·13·12·11·10· 9 · 8 · 7 · 6·5 · 4 ·3· 2 ·1 _
16·15· 14 · 13 ·1 2 ·11·1O·9· 8 ·7·6·5·4 .J.2 ·1
­

116,280.
Example 4: If26 people enter a 16-mile race and all of them finish, h w many
possi ble orders of fin ishing are there? P(26,26) = 26! 26! = 4.032 x 1026
D. Permutations th at contain repetitions or are placed in a Circular Pa ttern




8. Rational power: x· /b =
Exa m ple: x312 =

I

11.) I'
Il - r .

Exa mple I: How many ways can fi ve cartons of cereal be arranged?
P(5,5) ~ 5! = 5432 I = 120
Exa mple 2: If 18 people show up to serve on ajury, hO\ many 12 person
18!
P (1 8, 12)
juries
ca n
chose n?
be
( 18 -12 ) !

3. Distribution with multiplication : (xy)· = x·y·

Exam ple: (xy )s = x Sy 5

Exa m ple:

(

C. The notation n ! is read as n - factorial.

(::..64) = x lO

Exam ple: (

= !, P(n,r) =

x - 9 ) X 2 - 3x - 54

_(x 2 - 9x)

6x - 54

-(6x - 54)


o

Answer = x + 6 r. 0

2. Using synthetic division, we use the econd part of the divi or, but change
the sign (+9, trom the above example). We then Ii I the coefficients of the
div idend. From th e example above, th y \\ould be I, -3, -54.

Then bring down the coefficients one at a time, mUltiply them b} the


divisor (+9), and add them to the next coefficient. T his gives the same
result. From the example on pg. 3:

1-8
3
31

6. The product of a scalar (x), (a va lue with magn it ude, but no d irection),
and a matrix (.I) is d, with eac h e le ment f ..1 mu ltip lied time the

2..1 1 -3

-54

9 54

1 6 0 An wer = x + 6 r. 0

Note: The power of x a lways begins one lower in the answe r.

Exam ple 2: 3x3 + x 2 + 5x - 2 -;.- x + 1 -?


··
.
I 12 24 - 36 1

.

va I
ue x. S IX tIm es the matrI X .I =
1-1 8 0 12 1

7. The product of 2 - two by t\\'o matrices i a two b two matrix. The

::.113 1 5 -2

-3 2 -7

3 -2 7 -9
nswe r = 3x 2 - 2x + 7 r. -9

Examp le 3: In the follow ing example, noti ce that there mu t be a

place-ho lde r fo r missing powers of the variable. Z4 + Oz3 + Oz2 +

Oz + 16 -;.- z + 2


procedure isa fo llow: .J = 12
16
JxK='2 ( 6 ) + 4( 3 )

-21

0 0 0 16

-2 4 -8 16

I -2 4 -8 32 Answer = z3 _2z2 + 4z -8 r.32

C. S nthetic di vi ion of the form 3x2 + lOx - 9 + 3x~2. In thi example, all
nu mber are first divided by 3, then at the end of the process a ll fractions
are changed by mul tiplication.
Exam pie 1: 3x - 2 ) ( 3x 2 + lOx - 9) become x -

tl

t )(

X2

+

lj) x -

t


- 9

10

"3 3


8

2

3

3


4


t'

now mult ipl y a ll fract ions by 3.

Exam ple 2: 2x - I )( 6x 4 - x 3 - Ilx 2 + 9 x - 2), now div ide all parts
by 2, and use s nthetic divis ion .
3

1

II

9

2 -I

I

5


-5

2

-2-2
3

2 - 2


2
3

Add row 3 to

Aga in, when standard long di vision is perfor m d, the same answer is

atta ined . Note: II' there w re any frac ti ons in a ny part of the an wer,

they wou ld be removed by mul tip lication.


I I -2

I' W

I 1 -2 I : 7 I

107 0 :-7 1

10 0 ~ : 12 1


2

-2 1 I: 7 I

01 0: - I I

004 : 12 1

I -2 I : 7 1

o I 0: -I I

o 0 I: 3 1


Mu lt iply row 2 by 117

Multiply row .... by 1/4

MATRICES
A. Matrix:
rectangu lar array of elements in columns and rows. Rows are
named before columns, therefore a 2x4matri x has two rows and four colwnn .

Mu ltiply row 2 by 2 and add to row I

B. Prope r ties of Ma trices:
I. A ma trix with on l one row is called a row ma tri x.
2. matri th at ha onl. one o lum ll i ca lled a col um n mat rix .
3. Two matri ces are equal if and only if they have the same dimensions
and co ntai n the same identical e lements.
4. The sum o r a 2x3 and a 2x3 matrix is a 2x3 matrix in which the
e lements are added to the cor respond ing e leme nts.
Exam ple: Find.1 + K ifJ

= 1 2 4 -6 1and K =
1-3 0

f3


9

15
-3

21

Multiply row

1

2 4 -61
21

J

and K =


1-

-9

­ 21


I

3

II


1 00 : 2 1

00 1 : 3 1


The solution set = 2, -I, 3.


21

IS BN- 1 3 : 97 8 - 14 2320248- 6

IS BN- 10 : 142320 2 48 -1


-'i'

J~111 1

IlI1lllllllil1llillllil

U.S. 54.95 CAN . $7.50
Author: S. Orcutt

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5. The ditference of two matrices.J - K is equa l t add ing J to th e add itive

1- 3 0

by -I and add to r w I

I0 I: 5I

010: -I i

00 1 : 3 1


o I 0 : -I I


.J +KJ5 13 - 4 1.

12 - 3
II


inverse of K. .I =

26 1


6 ( 9 ) + l( 2 ) 1 139 56 1

[6 ( 6 ) + t( 3 )
C. sing matn ce to solve sys tems of eq uat ions: If you have thr e systems
of eq uation yo u can use an augmented matrix to find the solu tion et of
the va ria ble. You mu t fo llow the e guide li nes:
I. Any two rows Illa be interc hanged.
2. ny row may be repl aced by a non-zero multiple of that row.
3. Any row may be replaced by the um of that row and the multiple of

another. The goa l is to achieve an augmen ted matrix of the form;

110 0 : x,1

10 I 0 : y I

100 I : z I, where x, y, z = the olution el.

Example : Solve x - 2y + Z = 7

3x + Y - z = 2

2x + 3y + 2z = 7. u ing an augmented matrix.


Answer = 3x3 + x2 -5x + 2 r. 0

0

9 1.

21


2 (9) + 4( 2 )IJ 24

The aug mented matri i

Answer = x + 4 r. -1. When th i sa me problem is performed with trad i­
ti onal divi sion, the an wer i the same.

tl

41 ,1 6
I I 13

I: 7 I

13 I -I : 21

12 3 2: 71

Multiply row I by -3 and add 10 rO\ 2 I I -2 I: 7 I

107-4:-1 9 1

12 3 2: 7 1

MUltiply row I by -2 and add to I' w 3 I I -2 I: 7 I

10 7 -4 :-19 1

10 7 0: -7 1

Mu ltiply row 2 by - I and add trow 3 II -2 I: 7 1

I 0 7 -4 : -1 9 I

100 4 : 12 1


1

-3

Answer = x + 4 r. -

-5 - 81


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