47083207 Grade 11 Math Exam Notes

8/10/2019 47083207 Grade 11 Math Exam Notes

1/13

MCR3U Exam Review

Polynomials

A polynomial is an algebraic expression with real coefficients and non-negative integer exponents.

A polynomial with 1 term is called a monomial, x7 .

A polynomial with 2 terms is called a binomial, 93 2 x .A polynomial with 3 terms is called a trinomial, 973 2 + xx .

he degree of the polynomial is determined by the val!e of the highest exponent of the variable in the

polynomial.

e.g. 973 2 + xx , degree is 2.

"or polynomials with one variable, if the degree is #, then it is called a constant.$f the degree is 1, then it is called linear.

$f the degree is 2, then it is called %!adratic.

$f the degree is 3, then it is called c!bic.

&e can add and s!btract polynomials by collecting li'e terms.e.g. (implify.

( ) ( )

3)2)

*232*

*322*

*322*

23)

223))

23)2)

23)2)

++=

++=

++=

+

xxx

xxxxx

xxxxx

xxxxx

o m!ltiply polynomials, m!ltiply each term in the first polynomial by each term in the second.

e.g. +xpand and simplify.

( )( )

1272

12)32

32)

23)

223)

22

++=

+++=

++

xxxx

xxxxx

xxx

Factoring Polynomials

o expandmeans to write a prod!ct of polynomials as a s!m or a difference of terms.

o factormeans to write a s!m or a difference of terms as a prod!ct of polynomials.

"actoring is the inverse operation of expanding.

+xpanding

( ) ( ) 21*-7332 2 =+ xxxx"actoring

1

he negative in front of the brac'ets

applies to every term inside the brac'ets.

hat is, yo! m!ltiply each term by 1.

/rod!ct of

polynomials

(!m ordifference ofterms


2/13


3/13

MCR3U Exam Review

Rational Expressions

"or polynomialsFand G, a rational expression is formed when #, GG

F.

e.g.91)21

732 ++

+xx

x

%implifying Rational Expressionse.g. (implify and state the restrictions.

3,3

3

33

33

33

33

9-

92

2

+

=

+++=

+++=

++

mm

m

mm

mm

mm

mm

mm

m

Mltiplying and "ividing Rational Expressions

e.g. (implify and state the restrictions.

7,1,71

2

77

21

11

7

77

21

11

7

)91)

23

1

70

2

2

2

2

+

+=

++++

+

+=

++++

+

+=

++++

+

xxx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

x

xxA

3,1,),

1

3

31

1)

1)

33

31

1)

1)

33

1)

31

1)

33

)*

3)

)*

90

2

2

2

2

+=

++

+++

=

++

+++

=

++

+++

=

+++

++

x

x

x

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xx

xB

&dding and %btracting Rational Expressions

e.g. (implify and state the restrictions.

2,22

7*

22

1#*3

22

2*

22

3

2

*

22

3

2

*

)

30

2

+

=

+ +=

+

++

=

++

+=

++

xxx

x

xx

x

xx

x

xx

xxx

xxA

#,,#,

32

3

2

3

2

320

22

=

=

=

yyxyxxy

xy

yxxy

x

yxxy

y

yxyyxx

yxyxyxB


4/13

MCR3U Exam Review

Radicals

e.g. n a , is called the radical sign, nis the index of the radical, and ais called the radicand.

3 is said to be a radical of order 2. 3 is a radical of order 3.

=i'e radicals0 *3,*2,* >nli'e radicals0 3,*,* 3

+ntire radicals0 29,1-,

?ixed radicals0 7*,32,2)

A radical in simplest formmeets the following conditions0

"or a radical of order

n, the radicand has no

factor that is the nth

power of an integer.

he radicand contains

no fractions.

he radicand contains

no factors with

negative exponents.

he index of a radical

m!st be as small as

possible.

&ddition and %btraction of Radicals

o add or s!btract radicals, yo! add or s!btract the coefficients of each radical.

e.g. (implify.

( ) ( ) ( )

1#-311

1#-31*3)

1#2333*322

1#)339*3)2)#327*122

+=

+=

+=

+=+

Mltiplying Radicals

e.g. (implify.( )( ) ( )( ) ( )( ) ( )( ) ( )( )

( )

-1-

-2-312

3--2-32

333223233222332322

=

+=

+=

+=+

)

(ame order, li'eradicands

ifferent order ifferent radicands

22

22

2)

2

=

==


5/13

MCR3U Exam Review

Con'gates

( ( dcbadcba + and are called con@!gates.

&hen con@!gates are m!ltiplied the res!lt is a rational expression no radicals.

e.g. "ind the prod!ct.

( )( ) ( ) ( )

13

1*

29*

23*23*23* 22

===

=+

"ividing Radicals

Prime Factori(ation

"actor a n!mber into its prime

factors !sing the tree diagram

method.

Exponent Rles

Rle "escription Example

/rod!ct nmnm aaa += 7*2 ))) =!otient nmnm aaa = 22) *** =/ower of a power ( ) nmnm aa = ( ) )2 33 =/ower of a %!otient

#, =

bb

a

b

an

nn

*

**

)

3

)

3=

Bero as an exponent 1# =a 17# =


6/13

2

-2

MCR3U Exam Review

( ) ( )

1##

1

1#

1

1#

9133

2

2

222#

=

=

=

+=+

--

-

-2

232

-

23

232

3

3

)

2

2

22

ba

a

b

a

b

a

b

a

b

=

=

=

=

%olving Exponential E!ations

e.g. (olve forx.

22

2

2

2

99

19

739

739

=

=

+=

=

x

x

x

x

)

22

22

=+==

x

x

x

chec's)73

1

9

7392)

2

=====

==

xRS

RSLS x

Fnctions

A relationis a relationship between two sets. :elations can be described !sing0an e%!ation an arrow diagram a graph a table

73 2 = xy

in wordsDo!tp!t is three more than inp!tE

a set of ordered pairsF,),3,#,2,1G

f!nction notation

xxxf 3 2 =

he domainof a relation is the set of possible inp!t val!es xval!es.

he rangeis the set of possible o!tp!t val!es yval!es.

e.g. (tate the domain and range.

A0 F,),3,#,2,1G H0 60 *= xy omain G#, 1, )F

:ange G2, 3, F

Add to both sides.

(implify.


7/13

MCR3U Exam Review

A fnctionis a special type of relation in which every element of the domain corresponds to exactlyone element of the range.

7= xy and 1*2 += xy are examples of f!nctions. xy = is not a f!nction beca!se for everyval!e ofxthere are two val!es ofy.

he vertical line testis !sed to determine if a graph of a relation is a f!nction. $f a vertical line can be

passed along the entire length of the graph and it never to!ches more than one point at a time, then therelation is a f!nction.

e.g. A0 H0

)nverse Fnctions

he inverse, 1f , of a relation, f , maps each o!tp!t of the original relation bac' onto the

corresponding inp!t val!e. he domain of the inverse is the range of the f!nction, and the range of theinverse is the domain of the f!nction. hat is, if fba , , then 1, fab . he graph of

1 xfy = is the reflection of the graph xfy= in the line xy= .

e.g. ;iven*

13

= x

xf .

+val!ate 3f . +val!ate 123 +f

23

*

1#3

*

193

*

1333

=

=

=

=

f

f

f

f

)123

113

1*

*3

1

*

1-3

1*

1233123

=++=

+

=

+

=

+

=+

f

f

etermine 1 xf . +val!ate 21f

3

1*

3

1*

1*313*

*

13

*

13

1 +=

+=

+= =

=

=

xxf

xy

xyyx

yx

xy

3

112

3

11#

3

12*2

3

1*

1

1

1

=

+=

+=

+=

f

f

xxf

e.g. ('etch the graph of the inverse of the given f!nction xfy= .

7

2 2

his passes the

vertical linetest, so it is a

f!nction.

he line passes thro!gh more

than one point, so this relation

fails the vertical line test. $t isnot a f!nction.

:eplace all

xIs with 3.

+val!ate.

Mo! want to find the val!e of

the expression 123 +f .Mo! are not solving for 2f .

:ewrite xf as

*

13 =

xy $nterchangexand

y.

$f yo! have not already determined

1xf

do so.

>sing 1 xf , replace allxIs with

2.

4

2

-2

-4

2

-2

-

2

-2

-

raw the

lineyx.

:eflect thegraph in thelineyx.

xfy=

1xfy

=


8/13

4

-

-4

4

-

-4

2

-2

-

4

-2

-4

4

2

-2

-

MCR3U Exam Review

he inverse of a f!nction is not necessarily going to be a f!nction. $f yo! wo!ld li'e the inverse to alsobe a f!nction, yo! may have to restrict the domain or range of the original f!nction. "or the example

above, the inverse will only be a f!nction if we restrict the domain to F,#LG R xxx orF,#LG R xxx .

Transformations of Fnctions

o graph qpxkafy += 4 from the graph xfy= consider0

a determines the vertical stretch. he graph xfy= is stretched vertically by a factor of a. $f aN #then the graph is reflected in thex-axis, as well.

k determines the horiKontal stretch. he graph xfy= is stretched horiKontally by a factor ofk

1.

$f kN # then the graph is also reflected in the y-axis.

p determines the horiKontal translation. $fpO # the graph shifts to the right byp!nits. $fpN # then

the graph shifts left byp!nits.

q determines the vertical translation. $f qO # the graph shifts !p by q!nits. $f qN # then the graph

shifts down by q!nits.

&hen applying transformations to a graph the stretches and reflections sho!ld be

applied before any translations.

e.g. he graph of xfy= istransformed into

)23 = xfy . escribethe transformations.

"irst, factor inside the

brac'ets to determine theval!es of kandp.

( )( )

2,2,3

223

====pka

xfy

here is a vertical stretch of

3.

A horiKontal stretch of2

1.

he graph will be shifted 2

!nits to the right.

e.g. ;iven the graph ofxfy= s'etch the graph of

( )( ) 122 += xfy

xfy=

(tretch vertically

by a factor of 2.

:eflect iny-axis.

(hift to the

right by 2.(hift !p by 1.

his is the graph of

(( 22 += x


9/13

4

2

-2

-4

-5 5

aO#

minim!m

maxim!m

aN#

)2

71

=

+=

x

x

9)

33)

7)1))

==

=

f

f

f

*.22*.2

1#*.21#2*.22*.2

=+=

f

f

*adratic Fnctions

he graph of the %!adratic f!nction, cbxaxxf ++= 2 , is a parabola.&hen a> # the parabola opens !p. &hen a< # the parabola opens down.

+ertex Form: f x a x h k = +2

he vertex is , kh . he maxim!m or minim!m val!e is k.

he axis of symmetry isy h.

Factored Form: f x a x p x q = %tandard Form: f x ax bx c = + +2

he Keroes are px= and qx= . hey-intercept is c.

Complete t#e s!areto change the standard form to vertex form.

e.g.

( )( )( )

2*32

79232

7323-2

733-2

7-2

7122

2

2

222

222

2

2

++=

++=

+++=

+++=

++=

+=

xxf

xxf

xxxf

xxxf

xxxf

xxxf

Maximm and Minimm +ales

Pertex form, maxim!mQminim!m val!e is k.

"actored form0

e.g. etermine the maxim!m or minim!m val!e of 71 = xxxf .he Keroes of xf are e%!idistant from the axis of symmetry. he Keroes are 1=x and 7=x

.

he axis of symmetry isx ). he axis of symmetry passes thro!gh the vertex.hex-coordinate of the vertex is ). o find they-coordinate of the vertex,

eval!ate )f .

he vertex is 9,) . Heca!se ais positive 1=a , the graph opens!p.

he minim!m val!e is 9.

(tandard form0

e.g. etermine the maxim!m or minim!m val!e of 1#1#2 2 += xxxf witho!t completing the s%!are.

xxxg 1#2 2 = is a vertical translation of 1#1#2 2 += xxxf withy-intercept of #.*2 += xxxg *,# =x are the Keroes.

*.22

*# =

=x *.2=x is thex-coordinate of

vertex.

hey-coordinate of vertex

is 22.*. $t is a maxim!mbeca!se the graph opens down.

,eroes

"actor the coefficient ofx2form the terms withx2andx.

ivide the coefficient ofxby 2. (%!are this n!mber. Add and s!btract it.

Hring the last term inside the brac'et o!tside the brac'ets.

"actor the perfect s%!are trinomial inside the brac'ets.

(implify.

"actor xxxg 1#2 2 = to determinKeroes, then find the axis of symmetry. Hot

xf and xg will have the samex-

coordinates for the vertex. o find they-

coordinate forf(x)simply eval!atef(x)!sin

the samex-coordinate.


10/13

6

2

f x( )= 2x

6

2

f x( )= 2x-2+3

o determine the n!mber of Keroes of a %!adratic f!nction consider the form of the f!nction.

Pertex form0 $f aand khave opposite signs there are 2 Keroes 2 roots.$f a and khave the same sign there are no Keroes # roots.

$f k # there is one Kero 1 root.

"actored form0 f x a x p x q = - 2 Keroes. he Keroes are px= and qx= .2 pxaxf = - 1 Kero. he Kero isxp.

(tandard form0 6hec' discriminant. acbD )2 =$f #D there are 2 Keroes.

o determine the Keroes of from the standard form !se the !adratic formla.

"or , #2 =++ cbxax !sea

acbbx

2

)2 = to solve forx.

Reciprocal fnctions

he reciprocal f!nction of a f!nction, f , is defined as f1 . o help yo! graph 1xfy= , yo! sho!ld

!se the following0

he vertical asymptotes of

1

xfy= will occ!r where # =xf

As xf increases,

1

xfdecreases. As xf decreases,

1

xfincreases.

"or # >xf , #

1 >xf

. "or # se theinformation above to help yo! s'etch thereciprocal.

Exponential Fnctions

2

-2

-

5

2

-2

-

5

xxy )2=

xxy

)

1

2

=

Pertical asymptotes

$n general, the exponential f!nction

is defined by the e%!ation, xay= or

xaxf = , Rxa > ,# .

ransformations apply to

exponential f!nctions the same waythey do to all other f!nctions.


11/13

b a

b sinA

C

AB

Exponential .rowt# and "ecay/op!lation growth and radioactive decay can be modelled !sing exponential f!nctions.

;rowth0 dt

NtN 2 #= #N - initial amo!nt ecay0

h

t

NtN

=2

1 # #

N - initial amo!nt

t time elapsed t time elapsedd do!bling period h half-life

tN - amo!nt at time t tN - amo!nt at time

t

Compond )nterest

6alc!lating the f!t!re amo!nt0niPA 1 += A f!t!re amo!nt P present initial

amo!nt

6alc!lating the present amo!nt0 niAP += 1 i interest rate per conversion periodn n!mber of conversion periods

Trigonometry

;iven a right angle triangle we can !se the following ratios

Primary Trigonometric Ratios

r

y=sin

r

x=cos

x

y=tan

Reciprocal Trigonometric Ratios

sin

1csc ==

y

r

cos

1sec ==

x

r

tan

1cot ==

y

x

Trigonometry of /bli!e Triangles

%ine 0aw

c

B

b

A

a

sinsinsin==

6an be !sed when yo! 'now A(A, AA(, ((A

Cosine 0aw

Abccba cos2222

+=6an be !sed when yo! 'now (((, (A(

&hen yo! 'now ((A it is considered the ambig!o!s case.Angle 6onditions R of riangles

9# 2

r

y

x

A

b c

6 a H


12/13

1

0.5

-0.5

-1

50 100 150 200 250 300 350

1

0.5

-0.5

-1

50 100 150 200 250 300 350

5

-5

50 100 150 200 25 30 350

ba

C

AB

9#>A ba #ba> 1

Trigonometric )dentities

/ythagorean $dentity0 1cossin 22 =+ !otient $dentity0

cos

sintan =

e.g. /rove the identity. 222 cos1cos2sin =+

RS

LS

==

+=

++=

+=

2

2

222

22

cos

1cos1

1coscossin

1cos2sin

Periodic Fnctions

A periodic f!nction has a repeating pattern.

he cycleis the smallest complete repeating

pattern.

he axis of t#e crve is a horiKontal line that ismidway between the maxim!m and minim!m

val!es of the graph. he e%!ation is

2

min val!emax val!e +=y .

he periodis the length of the cycle.

he amplitdeis the magnit!de of the vertical

distance from the axis of the c!rve to the

maxim!m or minim!m val!e. he e%!ation is

2

min val!emax val!e=a

Trigonometric Fnctions

he graphs of sin=y , cos=y , and tan=y are shown below.sin=y tan=y

cos=y

Transformations of Trigonometric Fnctions

ransformations apply to trig f!nctions as they do to any other f!nction.

he graphs of dbkay ++= sin and dbkay ++= cos are transformations of the graphssin=y and cos=y respectively.

he val!e of adetermines the vertical stretch, called the amplitde.$t also tells whether the c!rve is reflected in the -axis.

sin=y/eriod 3#S

Amplit!de 1Beroes #S,

1#S, 3#ST

cos=y/eriod 3#SAmplit!de 1

Beroes 9#S, 27#ST

tan=y/eriod 1#S

Beroes #S, 1#S, 3#STPertical asymptotes 9#S, 27#ST

&or' with each side separately.=oo' for the %!otient or /ythagorean identities.

Mo! may need to factor, simplify or split terms !p.

&hen yo! are done, write a concl!ding statement.

(ince =(:( then 222 cos1cos2sin =+ is tr!e for all val!es of .


13/13

2

1.5

1

0.5

-0.5

-1

50 100 150 200 250 300 350 400

g x( )= cos 2x( )+1

f x( )= cos x( )

1

0.5

-0.5

-1

50 100 150 200 250 300 350 400

g x( )= 0.5sin x+45( )

f x( )= sin x( )

he val!e of kdetermines the horiKontal stretch. he graph is stretched by a factor ofk

1. &e can !se

this val!e to determine the periodof the transformation of sin=y or cos=y .

he period of ky sin= or ky cos= isk

3-#, k O #. he period of ky tan= is

k

1#, k O #.

he val!e of bdetermines the horiKontal translation, 'nown as the p#ase s#ift.he val!e of ddetermines the vertical translation. dy= is the e%!ation of the axis of t#e crve.

e.g. e.g.

12cos += y ( ))*sin2

1 += y

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47083207 Grade 11 Math Exam Notes