1

Section 1.1

GRAPHING CALCULATOR - WINDOW SIZING

WINDOW BUTTON.

Xmin=

Xmax=

Xscl=

Ymin=

Ymax=

Yscl=

Xres=resolution, smaller number= clearer graph

Larger number=quicker graphing

Xscl=5, Yscal=1 Xscl=10, Yscal=100

Xmin= , Xmax= Xmin= , Xmax=

Ymin= , Ymax= Ymin= , Ymax=

Set your viewing window to the following specifications and draw the results:

> @ > @30,30,10,1010,5

�� YsclXscl

> @ > @20,10,10,102,2

�� YsclXscl

-

Left point on × - axis -

Right point on × - ax , }

xmir|zy

. sal|

g× - mayDistancebetween tick - marks • •

w .

Bottom poihtony . ax , } Xsd-

Top point on y - ax , } =Distance between tick - marks •

% - mi,

!* 20 - 30 10- 2 z - 10

:- =

- =-# ⇒"=

2

4 3 10.2 2 02

x x x� � � 2 4 23 2 2 7 11 0x x x� � � �

RAPHING CALCULATOR - GRAPHING

STANDARD WINDOW

Graph each equation in a standard viewing window. " Zoom 6" gives you a standard window.

APPROPRIATE WINDOW

Graph the following , then 1) to change the window for the given interval.

1) Play with the ymin and ymax and the scalers. 2) "Zoom 0" will fit the graph within you xmin and xmax. 3) Use the table to find appropriate window limits.

5003002 �� xxy xy 2800�

FINDING AN UNKNOWN X -VALUE

Use the trace and zoom keys, then use the intersect key to find the matching coordinates.

xy 28� , � �0,x 532 �� xxy , � �9,x

POINT OF INTERSECTION

Graph the following and then use the intersect command to find the point of intersection.

xyxxy 512,102 � � 102.0,209 � � xyxy

SOLVING AN EQUATION

Use the intersect command to solve the equation

221

2.0 34 � � xxx 11722 243 2 �� � xxx

Use the zero command to solve the equation

(

x/Y÷€ X=1, -4

9=6+3×+5

C- 1.7,

20.9 ),(6.8/-21.9)

X=l .28

X= -

4:1=1.28

3

Section 1.2 FUNCTIONS

Ordered pair Relation Function Determine if the following are functions:

� � � � � �^ `3,2,1,2,0,1 �� � � � � � �^ `3,3,1,2,0,1 �

)(xf means the value of the function f at x . Be careful )(xf doesn’t mean f times x . x is also

called the "INPUT."

Let’s start with a function 42)( � xxf

)1(f

)5(f

)(f

)(hf

� )( hxf

� � � � � �^ `3,5,1,4,3,2 �

*tsetof ordered pairs { ( 1,234,16,D}A relation where for every input you get one and only

one output . Money (no repeating x - values )

function not a function function

ntttana.

Hit.

!!!!**§E±x+T functionnota function

2 ( l ) - 4 = - 226) -

4=6- ( 3+5 - ( o ) - 1

g( D= z g(o)= - 1

ZC } ) -4=25-4z C h ) - 4 = 2h - 4Zcxth) -4=2×+24-4

f- ( Q )=2( Q ) - 4

4

yxf )( � � � �1,2, oyx

Difference Quotient

Find for the following:

i) ii)

Functions Part II

52 �� xy 52)( �� xxf

� � 1522)2( �� f 1) Write as a function. Plug it into your calculator and find f(-3). 2) Given 42)( � xxg , Find: a) g(2) b) x when g(x) = 5 c) Fill in the blank (___, 4) 3) A rental company's daily charges are calculated using the function , where x is the number of miles. Translate this algebraic statement into a verbal statement explaining the rental companies daily charges.

( HDCXTD

pfc#⇒fftthkI

, Oifcxth's.es#yyQtyhfyn(3x+3hyDt3x-4)_20 µ¥

�2� -35×+34-+44�3�2×h+hy-=

4 =D Zxth -4

.

FCD - 5×-4 FH )= -199

y=z×-y y=S

4$94=214.4

5=2×-44--2×-4=4.4+4 +4 +4 +4

gH=o ¥¥8=2×11--4.504¥

Xis#of milesCCD is Avental company's daily charges

A rental companies daily charges are $45 pluszo¢ per mile

.

5

Domain and Range Domain: Range: Determine the domain and range for the following:

� � � � � �^ `3,2,1,2,0,1 �� � � � � � �^ `3,3,1,2,0,1 �

221�

x

y 1353

��

xx

y 2xy

4� xy 32 �� xy 2

2�

xx

y

� � � � � �^ `3,5,1,4,3,2 �

× . values y . value ,

C inputs ( oultputs)

D :{2,4/53 D :{ -1,23R :{ 3

,-1 } R :{ 0

, -1,33

Fraction :

Denonto

2×-2-+0 3×+1*0 R2×1=2 3×1=-1xtl xtt

lrexceptl Bnexceptstto ,DV(1,0 too ,±DVt÷,o)

F=h±al

Fgsoreal - zx . }zo ×

-2=10×20×+420. zxzz Xtz

Xz - y #*-

- 3- ¥omxEgeriff E •,

' ⇒ [0 ,2)v(3a)#u

6

GRAPHS - PROPERTIES Section 1.3 Find the following: A) DOMAIN B) RANGE C) X-INTERCEPTS D) Y-INTERCEPTS E) INTERVALS OF INCREASING F) INTERVALS OF DECREASING G)Local Extrema a) c) e) b) d) f) g) a) a)

b) b)

c) c)

d) d)

e) e)

f) f)

g) g)

-

96)

( 2,4):

↳ •••• "}y,;D'Es;:3 ¥

'¥5"'"

' 2

¥B¥I÷Yw

,

�2�

T| �3�A÷K¥#,

'Fas,tDv¥d¥±iY%2

Fines

ESFD Eo%- )( i

,- D none

7

Using your calculator draw a quick sketch and mark any points of interest you can find, then find the intervals or points for the following: a) domain b) range c) x-intercepts (calc.) d) y-intercepts (calc.) e) intervals of increasing f) intervals of decreasing g) local extrema h) domain and range

52)( �� xxf 5.15.125.005.0)( 23 ��� xxxxm

a) a) b) b) c) c) d) d) e) e) f) f) g) g) h) h) DISCONTINUITY Breaks of gaps in the graph. Find the domain and locate any points of discontinuity.

to5.36.03 )

9

rate .

G ;t.fotew.pe#

,

[ 2,E§ %9, -3.1 )

a

C- a ,A ) a as ,d )C- 5,8 ) C- a ,a )-7

,3 -8.2

, -0.89,

4.1

¥ is ) EE,

- s .D ,[ 1.9 ,A )fcs , -2 ] [-5.2/1,9]4- s ) find;D ,auaE'D

--

×+2=tO. .

X=tz÷.

0 ::

:

Difo,

- Dvtzcs ) Diego )poihtofd , }c

.at Points of disc

.

atX= -2 X= 0.5k , whereKisaunint

-

8

Piece-Wise Graphs

� � � � � � � � � � � �2

3 1 if 1

Graph the function if 1 4 , and evaluate 3 , 1 , 0 , 4 , 75 if 4

x x

f x x x f f f f fx x

� d �­° � � � � �®°� � t¯

APPLICATIONS

i) The profit (in dollars) from the sale of x car seats for infants is given by Find the number of car seats that must be sold to maximize the profit. What is the maximum profit (to the nearest dollar)? ii) A box with no top is to be made from a 10 by 20 inch cardboard by cutting equal size squares from each

corner and folding up the sides. Let x be the length of the side of the square to be cut from each corner. Answer the following:

a) What is the restriction on x?

b) Find the value of x that will maximize the volume of the box. What is the maximum volume?

c) Find the value of x at which the volume of the box will be greater than 150 cubic inches?

x y x y x y

o

YIKH Itau "

EI¥••n¥oy;.

:S to

%.

.y0 4 16•.

%

X= 543 car seatsC 542.86

,

6314.29£< S lot

20 -:0

V =L . W . H = ( zo - 2×340 - 2×)×

( 2.11,192,45 ) x= 2.11 rn

V= 192^45 in'

-

(1.07in,3.305in )

9

Section 1.4

GRAPHS AND TRANSFORMATIONS

PARENT GRAPHS

Sketch the shape of each in the box below. There is no grid. I just want the shape.

� � 12 � xxf � � 2xxf � � 3xxf � � xxf � � 3 xxf � � xxf

VERTICAL SHIFTS Graph the following and look at the parent graphs to compare.

� � 2� xxf � � 32 � xxf � � 1� xxf

What is happening to the graph of the original function when you add or subtract to the original function?

� � cxfy � � � cxfy �

HORIZONTAL SHIFTS Graph the following and look at the parent graphs to compare.

� � 1� xxf � � � �22� xxf � � 1� xxf

What is happening to the graph of the original function when you add or subtract to the x to the original function?

� �cxfy � � �cxfy �

ttf. em

shifts shifts

1k " units"

(" units

up dowy

"

(" in ,B "

c"

left unitnight

10

Graph without using a calculator, just move the original function.

� � 1� xxf � � � �22� xxf � � 1� xxf

� � xxf � 1 � � � � 32 � xxf � � xxf �� 2

� � 11 �� xxf � � � � 32 2 �� xxf � � 12 ��� xxf

Give the function of the following: The graph is shifted four units to the left and five units down.

( yo ) (2/0)

4,01µ Y as

Co,, ) (9-3) copy

µ µ .

( 1,1 ) (3-3) fl , -2 )

µ µ 9

:OC-6,3)

rs , ,fcxtktyp .g

Y=1×+61t3 y=k -55.3

11

� �31

3f x x§ · ¨ ¸

© ¹

REFLECTION ABOUT THE X-AXIS Graph the following and look at the parent graphs to compare.

� � xxf � � � 2xxf � � � 3xxf �

What is happening to the graph of the original function ? REFLECTION ABOUT THE Y-AXIS Graph the following and look at the parent graphs to compare.

� � xxf � � � 3 xxf � � � � �3f x x �

What is happening to the graph of the original function when you multiply a negative to the original function? STRETCHING AND SHRINKING GRAPHS VERTICAL STRETCHING AND SHRINKING GRAPHS Graph the following and look at the parent graphs to compare.

� � xxf 2 � � 23xxf � � xxf 5.0 � � 33/1 xxf �

� �0.5 f x

HORIZONTALY STRETCHING AND SHRINKING GRAPHS Graph the following and look at the parent graphs to compare. � � 2f x x � � � �23f x x � � 0.5f x x

� �2f x � �0.5f x

Find: 0)=

� �2 f x

reflected about the × - axB

reflects about they - axis

vertically stretched vertically Shrunkby a Factor of 2 by a Facto 0 . g

ZFCX)

horizontally shrunk horizontallyby a Factual 2 Stretched by a facto -

of

0i= -37.5 , 1.4

nult . "y"by 2 . -2,2C. 4/4 .com yen "

tri"

" III'm'm"IYK

"

" #C- 2

,.2) ( 2

,- 2)

FCX )+3 FCX - 3)

12

PUTING IT ALL TOGETHER

( )af x h k� �

If a is negative = h= k= Graph without using a calculator, just move the original function.

� � 1�� xxf � � � �2221 �� xxf � � xxf � 2 � � � �2121

�� xxf

Find an equation for the function g. Check your work with a calculator. The graph of is shifted five units to the right and four units up. The graph of is vertically stretched by a factor of 2, reflected in the x-axis, and then shifted 6 units up. The graph of is shifted 6 units up, vertically stretched by a factor of 2, and then reflected in the x-axis. The graph of is shifted to the left 3, vertically stretched by a factor of 0.5, and then shifted 2 units down. The graph of is shifted 2 units down, vertically shrunk by a factor of 0.5, and then shifted to the left 3. The graph of is shifted 4 units up, vertically shrunk by a factor of 1/3, shifted to the right 1 unit, and then reflected in the x-axis.

( h ,K) staAnarrow wide ( flips )

Vert,

vent . open ,Stretch Shark douy

shift left shift uponor right down

to +0a- -1 4,0 ) a=2 ( 2,1 ) ( 0,2 ) ( 1,0)

geog ^¥he pay

Y= 335*4+2

gcx )= &- s ) 3+4

gcx )= .2rx+6

91×1=-5×+6)#cK

gcx )=0.KXt3M - 2

g(x)=%x+35- D

0 :sE2+(×+3D

faith-11+4 ]

13

EVEN-ODD FUNCTIONS EVEN FUNCTION-- � � � �xfxf � --so symmetric about the y-axis Why are the two functions equal?

a) � � 2xxf and � � � �2xxg � b) � � xxf and � � xxg �

ODD FUNCTION-- � � � �xfxf � � --so symmetric about the origin Why are the two functions equal?

a) � � 3xxf � and � � � �3xxg � b) � � 5 xxf � and � � 5 xxg �

Prove analytically that the following function is even or symmetric about the y-axis:

� � 32 � xxf � � 42 �� xxxf

Start with: � � 32 � xxf

� � � �2 3f x x� � �

Prove analytically that the following function is odd or symmetric about the origin:

� � xxxxf 32 35 ��� � �x

xf21

Start with: 1) � � � �5 32 3f x x x x� � � � �

2) � � � � � � � �5 32 3f x x x x� � � � � � �

ftxhfxjhttxtyffx)=X23 FH Xhtktyftp.xz.sfsane ,

tcx}Ex ' +1×1+4even fund

. even fund.

XS -2×3+3 ×- ftp.#=ztx

ftD=×s-z×3+3×fl '×)=tEy= - ztx

Ffx )= - fcx ) ffX)==fC×)oddfuncf

. dddfnwcsl .

14

Section 1.5

OPERATIONS ON FUNCTIONS

� �� � � � � �xgxfxgf � � � �� � � � � �xgxfxgf � � � �)()(xgxf

xgf

¸̧¹

·¨̈©

§

42)( � xxf , 42)( 2 � xxg

Find:

� �� �xgf �

� �� �xgf �

� �� �3gf �

� �� �xfg

� �xfg¸̧¹

·¨̈©

§

FIND THE DOMAIN OF

� �� � � � � �xgxfxgf � � � �� � � � � �xgxfxgf � �

� � 3� xxf � � 4� xxg

� �)()(xgxf

xgf

¸̧¹

·¨̈©

§

� �� � � � � �xgxfxfg �

� �� � � � � �xgxfxfg �

= (2×-4)+(2×24)=2×2+2×-8= (2×-4)-(2×44)=2×-4 . 2×74=2×2+211

= FG ) - gG)=2- 14=-12 F-g)1D=[2,62+4*34= (2×-4) (2×2-4)=4×3-8×2- 8×+16

iYx÷n#¥=x÷D2

ftgkxtxttrxyfk )

penny(f- g)CD=FtFy

||/#H$FokxtxrxryYK) fund-4 X-

3>-0×+420×23×2- 4together §M[},§

CoyX -

320×+4>0×23× > - 4

ETCy [3* )

15

COMPOSITE FUNCTIONS � �� � � �� �xgfxgf $

Given: 4)( 2 � xxf , 43)( � xxg , � � 24 xxp �

Find: � �� �xfg $

� �� �xpf $

� �� �2pg $

Find � �� �2gf $

� �� �4gf $

� �� �3fg $

� �� �1fg $

� �� �3gf � � �� �2f g �

� �� �4f g �

� �� �0g f

� �� �1fg $

� �� �1f g�

� �� �0f g�

� �� �0f gx

Find a functions f(x) and g(x) such that the

( popopopj( 2)

f.. gffcx))=g(×t4)⇒(x2y) - 43×2-12-4

GCFKH 3×-16

⇐ =f(#n=(Fat4=4×2-4o 2 fop)(× ) = - ×2

pCz)=Fap=ogco )=3( o ) -4 §(fogop)( 0 ) = pco )=2 g(z)=z f(z)=@

= 1

=3= 4

= 2

=- ,

FG ) . g(})2:HTIEYSIY

,fH*o9,42) =ffy)=t£g) p ) = g(gfD)= - 4

= g(ff( p ) = - 2

= FCD - ga )=-3

- ( 61=-9=f( of 1-g(D= 0+2=2

=f( D .g(o)= -0.2=0

0.94=1×+15-3

|

gcx )=(×tD3 9KI=x+i

fW=r× f(× ) =¥ Fatma

16

DOMAIN OF COMPOSITE FUNCTIONS

Remember, we are looking for the real number values of x that we can input an get real number outputs.

4)( 2 � xxf , )(xg x�4

Find the domain of composite functions f(g(x))

1) f(g(x))

Find the domain of composite functions g(f(x))

1) g(f(x))

Find the domain of composite functions f(g(x))

1) f(g(x))

Find the domain of composite functions g(f(x))

1) g(f(x))

FGW )=ft4¥4gcxt.FI Asa ,,=4Yj44- Xzo R¥xE¥

t94##=y÷xzCfcxj) 8×40

9- 8- xto1/3 Ezogzod ±T=#

EZFG zmzy ±Zrz=×

fG#*##*jtxto2×+4go.IE#*t*FxxIE,¥¥.×¥,

xtz Xt3

Rexcept -2kC- s,:3 )Vf3

,

- DUH ,a)

##µ¥s*x!¥IEtEI?TaX-D X=Z F÷4Rexcept 2,3 C- p/BvGDv4D Ia ,=*±z

17

Section 1.6 ONE-TO-ONE FUNCTIONS A function in which each element in the range corresponds to one and only one element in the domain. Determine if the following are One-to-one functions:

� � � � � �^ `3,2,1,2,0,1 �� � � � � � �^ `3,3,1,2,0,1 �

If implies , then is one-to-one Determine if the following are One-to-one functions:

23 � xy 32 � xy

� � � � � �^ `3,5,1,4,3,2 �

function hotafunct.

functionone - to - one

one - to - one function not a fnnctsy

I.:b.#E÷tT.tn#tFfW=3Xt2F(x)=Xt3f(a)=3at2fk)=a2

- 3

fcb ) =3btZ f1b)= b2 - 3f ( a )=fCb) AZ . 3=42-33a#=3bII F2=bT3#=¥ at±b

a=b ✓ hot one - to - one

18

INVERSE FUNCTIONS

If is one-to-one, then an Inverse function , , exists Find the inverse by switching the x and the y of the following:

� � � � � �^ `3,3,1,2,0,1 �

Find the inverse of the following: Ex/ 23 � xy Ex/ 33 � xy

2� xy , Ex/ xx

y3�

Finding the inverse of a graph:

2)( � xxf

� � � � � �^ `4,5,1,4,3,2 �

{(3/4,44)/(4/5)}fw=3Xtz GG)

X=3yt2 X=y3 . )

'#¥ Yxtzy3 Txt =y

¥=Y

ftp.fxF 'Cx)= '€

D ! Xto( Dttfytzpriiyz-2 D :×20 ( y

)×=Y-3#Xkytz y

y=×t2 Xy=y - 3

hTX)=×t2 DIXZO xy . y= -3

YC × - D= - 3* ,

hcxtxz f- ' ( × ) :# ,Dixtl

a*¥I,

- 2)

19

INVERSE FUNCTIONS - DETERMINING IF A FUNCTION IS AN INVERSE

� �� � xxff �1$ for domain of � �xf 1� , and � �� � xxff � $1 for domain of � �xf

Determine

are inverses. Determine are inverses. Why are we looking for x? = Why do you need to check both? APPLICATIONS OF INVERSES The number q of cd players a retail chain is willing to supply at a price of $p is given approximately by

a) Find the range of S using your calculator. b) Find , and find it's domain and range. HINT: DO NOT SWAP THE LETTERS, THIS WILL CAUSE CONFUSION.

fcscxt.ae#+s=x*+X=xVgCf/N) -3×424= 3¥ = × ✓

fcgcxj-xf.sk#=c**jE*=xrx+t-x=xrff1)=

1f ( 14 ) = 4

f(r÷- ( 2 )= 4 g(D= I

300 ±qE 700

300 ES E70o

Cphgqi.gg#o_CptXDp=g2o0gq30oaqE7ooqPt2oq=9oop5'Cq)=2go0Tg2oq=9oop-qP.20q=pGoo

-

q )