mcr3u exam review package ch. 1: functionsgiangioppo-ba-mathematics.weebly.com/uploads/4/6/7/... ·...

MCR3U – Exam Review Package

2

-2

4

2

Ch. 1: Functions A relation is a relationship between two sets. Relations can be described using: an equation an arrow diagram a graph a table

73 2 � xy

in words “output is three more than input”

a set of ordered pairs function notation

)}8,4(),3,0(),2,1{( xxxf 3)( 2 � The domain of a relation is the set of possible input values (x values). The range is the set of possible output values (y values). e.g. State the domain and range. A: )}8,4(),3,0(),2,1{( B: C: 5� xy Domain = {0, 1, 4} Range = {2, 3, 8} A function is a special type of relation in which every element of the domain corresponds to exactly one element of the range.

7� xy and 152 � xy are examples of functions. xy r is not a function because for every value of x there are two values of y. The vertical line test is used to determine if a graph of a relation is a function. If a vertical line can be passed along the entire length of the graph and it never touches more than one point at a time, then the relation is a function. e.g. A: B:

x y 1 2 2 3 3 4 4 3

8 -1 0 7 6 -3 3 -5

g

4

2

4

2

What value of x will make x – 5 = 0? x = 5 The radicand cannot be less than zero, so Domain = },5|{ R�t xxx Range = },0|{ R�t yyy

This passes the vertical line test, so it is a function.

The line passes through more than one point, so this relation fails the vertical line test. It is not a function.


Inverse Functions The inverse, 1�f , of a relation, f , maps each output of the original relation back onto the corresponding input value. The domain of the inverse is the range of the function, and the range of the inverse is the domain of the function. That is, if fba �),( , then 1),( �� fab . The graph of )(1 xfy � is the reflection of the graph

)(xfy in the line xy .

e.g. Given 5

13)( �

xxf .

Evaluate )3(�f . Evaluate 1)2(3 �f

2)3(510)3(

519)3(

51)3(3)3(

� �

� �

��

��

f

f

f

f

41)2(31)1(3

1553

15

163

15

1)2(331)2(3

��

�¸¹·

¨©§

�»¼º

«¬ª �

�»¼º

«¬ª �

�

f

f

Determine )(1 xf � . Evaluate )2(1�f

315)(

315153135

513

513

1 � ?

�

� �

�

�

� xxf

xy

xyyx

yx

xy

311)2(

3110

31)2(5)2(

315)(

1

1

1

�

�

�

�

�

�

f

f

xxf

e.g. Sketch the graph of the inverse of the given function )(xfy . The inverse of a function is not necessarily going to be a function. If you would like the inverse to also be a function, you may have to restrict the domain or range of the original function. For the example above, the inverse will only be a function if we restrict the domain to },0|{ R�t xxx or },0|{ R�d xxx .

Replace all x’s with –3. Evaluate.

You want to find the value of the expression 1)2(3 �f . You are not solving for )2(f .

Rewrite )(xf as 5

13 �

xy

Interchange x and y. Solve for y.

If you have not already determined )(1 xf � do so.

Using )(1 xf � , replace all x’s with 2. Evaluate.

4

2

-2

-4

4

2

-2

-4

4

2

-2

-4

Draw the line y = x.

Reflect the graph in the line y = x.

)(xfy

)(1 xfy �


4

2

-2

-4

4

2

-2

-4

4

2

-2

-4

4

2

-2

-4

4

2

-2

-4

Transformations of Functions To graph cdxkafy �� )]([ from the graph )(xfy consider:

a – determines the vertical stretch. The graph )(xfy is stretched/compressed vertically by a factor of a. If a < 0 then the graph is reflected in the x-axis, as well.

k – determines the horizontal stretch. The graph )(xfy is stretched/compressed horizontally by a factor of

k1 . If k < 0 then the graph is also reflected in the y-axis.

d – determines the horizontal translation. If d > 0 the graph shifts to the right by d units. If d < 0 then the graph shifts left by d units.

c – determines the vertical translation. If c > 0 the graph shifts up by c units. If c < 0 then the graph shifts down by c units.

When applying transformations to a graph the stretches and reflections should be applied before any translations. e.g. The graph of )(xfy is transformed into )42(3 � xfy . Describe the transformations.

First, factor to determine the values of k and d.

� �� 2,2,3

223

� dka

xfy

There is a vertical stretch of 3, a

horizontal compression of 21 , and

a horizontal translation 2 units to the right.

e.g. Given the graph of )(xfy sketch the graph of � �� 122 �� xfy

)(xfy Stretch vertically by a factor of 2.

Reflect in y-axis.

Shift to the right by 2.

Shift up by 1.

This is the graph of � �� 122 �� xfy


Ch. 2: Algebraic Expressions 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 . The degree of the polynomial is determined by the value of the highest exponent of the variable in the polynomial. e.g. 973 2 �� xx , degree is 2. For polynomials with one variable, if the degree is 0, then it is called a constant. If the degree is 1, then it is called linear. If the degree is 2, then it is called quadratic. If the degree is 3, then it is called cubic. We can add and subtract polynomials by collecting like terms. e.g. Simplify.

� � � �

3424523255322553225

234

22344

23424

23424

��

��

��

��

xxxxxxxx

xxxxxxxxxx

To multiply polynomials, multiply each term in the first polynomial by each term in the second. e.g. Expand and simplify.

� ��

12872128432

324

234

2234

22

��

��

��

xxxxxxxxx

xxx

Factoring Polynomials To expand means to write a product of polynomials as a sum or a difference of terms. To factor means to write a sum or a difference of terms as a product of polynomials. Factoring is the inverse operation of expanding. Expanding Æ � �� 21567332 2 �� xxxx Å Factoring

The negative in front of the brackets applies to every term inside the brackets. That is, you multiply each term by –1.

Product of polynomials

Sum or difference of terms


Types of factoring: Common Factors: factors that are common among each term. e.g. Factor,

� �8357562135

22

22233

��

��

nmnnmnmnmnm

Factor by grouping: group terms to help in the factoring process. e.g. Factor,

� � � ��

� �� nmyxnmynmxmnynmxmynynxmx

mynxnymxA

��

��

444

4444:

)231)(231(

]2)31][(2)31[(4)31(

4961:22

22

yxyxyxyx

yxyxxB

��

��

��

Factoring cbxax ��2 Find the product of ac. Find two numbers that multiply to ac and add to b. e.g. Factor,

)7)(2(149: 2

�� yy

yyA

)3)(23()3(2)3(3

6293673:

22

22

yxyxyxyyxx

yxyxyxyxyxB

��

��

��

Sometimes polynomials can be factored using special patterns. Perfect square trinomial ))((2 22 babababa �� or ))((2 22 babababa �� e.g. Factor,

2

2

)32(9124:

�

��

pppA

)25)(25(4

)42025(41680100:

22

22

yxyxyxyx

yxyxB

��

��

Difference of squares ))((22 bababa �� e.g. Factor, )23)(23(49 22 yxyxyx �� Things to think about when factoring:

x Is there a common factor? x Can I factor by grouping? x Are there any special patterns? x Check, can I factor cbxx ��2 ? x Check, can I factor cbxax ��2 ?

Product = 14 = 2(7) Sum = 9 = 2 + 7

Product = 3(–6) = –18 = –9(2) Sum = – 7 = –9 + 2 Decompose middle term –7xy into –9xy + 2xy. Factor by grouping.

Each term is divisible by nm27 .

1+6x+9x2 is a perfect square trinomial

Difference of squares

Common factor

Recall n – m = –(m – n)

Group 4mx – 4nx and ny – my, factor each group


Rational Expressions

For polynomials F and G, a rational expression is formed when 0, zGGF .

e.g. 91421

732 ��

�xx

x

Simplifying Rational Expressions e.g. Simplify and state the restrictions.

3,33

)3)(3()3)(3()3)(3()3)(3(

969

2

2

�z��

��

��

��

�

mmm

mmmmmmmm

mmm

Multiplying and Dividing Rational Expressions e.g. Simplify and state the restrictions.

7,1,)7)(1(

)2()7)(7()2)(1(

)1)(1()7(

)7)(7()2)(1(

)1)(1()7(

491423

17: 2

2

2

2

�rz��

�

��

u��

�

��

u��

�

��

u��

xxx

xxxxxx

xxxx

xxxx

xxxx

xxxx

xxxA

3,1,4,)1()3(

)3)(1()1)(4(

)1)(4()3)(3(

)3)(1()1)(4(

)1)(4()3)(3(

)1)(4()3)(1(

)1)(4()3)(3(

4534

459: 2

2

2

2

r�z��

��

u��

��

u��

��

y��

��

y��

�

xxx

xxxx

xxxx

xxxx

xxxx

xxxx

xxxx

xxxx

xxxB

Adding and Subtracting Rational Expressions e.g. Simplify and state the restrictions.

2,)2)(2(

75)2)(2(

1053)2)(2(

)2(5)2)(2(

32

5)2)(2(

32

54

3: 2

rz��

�

��

��

��

��

��

��

�

xxx

xxx

xxx

xxx

xxx

xxA

0,,0,)(

32)(

3)(

2)(

3)(

2

32: 22

zz��

��

�

��

�

��

�

yyxyxxyxy

yxxyx

yxxyy

yxyyxx

yxyxyxB

Note that after addition or subtraction it may be possible to factor the numerator and simplify the expression further. Always reduce the answer to lowest terms.

Factor the numerator and denominator. Note the restrictions. 3�zm

Simplify.

State the restrictions.

Factor. Note restrictions. Simplify.

State restrictions.

Factor. Note restrictions. Invert and multiply. Note any new restrictions. Simplify.

State restrictions.

Factor. Note restrictions. Simplify if possible. Find LCD.

Write all terms using LCD.

Subtract. State restrictions.

Factor. Note restrictions. Simplify if possible.

Find LCD. Write all terms using LCD.

State restrictions.

Add.


4

2

-2

-4

-5 5

a>0

minimum maximum

a<0

42

71

�

x

x

9)4()3(3)4(

)74)(14()4(

� �

��

fff

Ch. 3: Quadratic Functions The graph of the quadratic function, cbxaxxf �� 2)( , is a parabola. When a ! 0 the parabola opens up. When a � 0 the parabola opens down. Vertex Form: f x a x h k( ) ( ) � �2

The vertex is ),( kh . The maximum or minimum value is k. The axis of symmetry is y = h.

Factored Form: ))(()( sxrxaxf �� The zeroes are rx and sx .

Standard Form: f x ax bx c( ) � �2

The y-intercept is c. Complete the square to change the standard form to vertex form. e.g.

� ��

25)3(2)(7)9(2)3(2)(

7)3(2362)(73362)(

762)(7122)(

2

2

222

222

2

2

��

��

��

��

��

��

xxfxxf

xxxfxxxfxxxfxxxf

Maximum and Minimum Values Vertex form, maximum/minimum value is k.

Factored form: e.g. Determine the maximum or minimum value of )7)(1()( �� xxxf . The zeroes of )(xf are equidistant from the axis of symmetry. The zeroes are 1 x and 7 x .

The axis of symmetry is x = 4. The axis of symmetry passes through the vertex. The x-coordinate of the vertex is 4. To find the y-coordinate of the vertex, evaluate )4(f .

The vertex is )9,4( � . Because a is positive ( 1 a ), the graph opens up. The minimum value is –9.

Factor the coefficient of x2 form the terms with x2 and x.

Divide the coefficient of x by 2. Square this number. Add and subtract it.

Bring the last term inside the bracket outside the brackets.

Factor the perfect square trinomial inside the brackets.

Simplify.


Zeroes To determine the number of zeroes of a quadratic function consider the form of the function. Vertex form: If a and k have opposite signs there are 2 zeroes (2 roots). If a and k have the same sign there are no zeroes (0 roots). If k = 0 there is one zero (1 root). Factored form: ))(()( sxrxaxf �� → 2 zeroes. The zeroes are rx and sx . 2)()( rxaxf � → 1 zero. The zero is x = r. Standard form: Check discriminant. acbD 42 � If 0�D there are no zeroes. If 0 D there is 1 zero. If 0!D there are 2 zeroes. To determine the zeroes of from the standard form use the quadratic formula.

For 02 �� cbxax use a

acbbx2

42 �r� to solve for x.

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

3 is said to be a radical of order 2. 3 8 is a radical of order 3. Like radicals: 53,52,5 � Unlike radicals: 3,5,5 3

Entire radicals: 29,16,8 Mixed radicals: 75,32,24 A radical in simplest form meets the following conditions:

For a radical of order n, the radicand has no factor that is the nth power of an integer.

The radicand contains no fractions.

The radicand contains no factors with negative exponents.

The index of a radical must be as small as possible.

Same order, like radicands Different order Different radicands

22

22

2482

u

u

Not simplest form

Simplest form

262626

22

23

23

2

2

u

Simplest form

aaaa

aa

a

aa

u

�

2

1

1

1

Simplest form

3

33 24 2

Simplest form


Addition and Subtraction of Radicals To add or subtract radicals, you add or subtract the coefficients of each radical. e.g. Simplify.

� � � � � �

106311

10631534

1023335322

1043395342403275122

��

��

��

u�u�u ��

Multiplying Radicals

e.g. Simplify. � ��

� �

616

6263182

3662632

333223233222332322

��

��

��

��

Conjugates

� � � �dcbadcba �� and are called conjugates. When conjugates are multiplied the result is a rational expression (no radicals). e.g. Find the product. � ��

13185

)2(95235235235

22

� � �

� ��

Dividing Radicals

Express each radical in simplest form.

Collect like radicals. Add and subtract.

Use the distributive property to expand

Multiply coefficients together. Multiply radicands together.

Collect like terms. Express in simplest form.

Opposite signs

Same terms Same terms

e.g. Simplify.

63225

3035

102

5303

5102

5303102

�

�

� �

0,0,,, tt� bababa

ba R

0,0, tt u baabba


6

4

2

f x� � = 2x6

4

2

f x� � = 2x-2+3

Ch. 4: Exponential Functions Exponent Rules Rule Description Example Product nmnm aaa � u 752 444 u Quotient nmnm aaa � y 224 555 y Power of a power � � nmnm aa u � � 842 33

Power of a quotient 0, z ¸¹·

¨©§ b

ba

ba

n

nn

5

55

43

43

¸¹·

¨©§

Zero as an exponent 10 a 170

Negative exponents 0,1z � a

aa m

m 22

919 �

Rational Exponents � �mnn mnm

aaa � �433 434

272727 e.g. Evaluate. e.g. Simplify.

� � � �

1001

101

109133

2

2

2220

� ��

��

66

6

62

)2(32

6

23

)2(32

3

3

4

22

)2(2

ba

abab

ab

ab

¸̧¹

·¨̈©

§

�

��

�

��

��

�

Exponential Functions Exponential Growth and Decay Population growth and radioactive decay can be modelled using exponential functions.

Growth: dt

NtN )2()( 0 0N - initial amount Decay:ht

NtN ¸¹·

¨©§

21)( 0 0N - initial amount

t – time elapsed t – time elapsed d – doubling period h – half-life )(tN - amount at time t )(tN - amount at time t

Follow the order of operations. Evaluate brackets first.

Power of a quotient.

Power of a product.

In general, the exponential function is defined by the equation, xay or xaxf )( , Rxa �! ,0 . Transformations apply to exponential functions the same way they do to all other functions.


b a

b sinA

C

A B

ba

C

A B

Ch. 5: Trigonometry Given a right angle triangle we can use the following ratios Primary Trigonometric Ratios

ry

Tsin rx

Tcos xy

Ttan

Reciprocal Trigonometric Ratios

TT

sin1csc

yr

TT

cos1sec

xr

TT

tan1cot

yx

Trigonometry of Oblique Triangles Sine Law

Cc

Bb

Aa

sinsinsin

Can be used when you know ASA, AAS, SSA Cosine Law

Abccba cos2222 �� Can be used when you know SSS, SAS When you know SSA in Sine Law it is considered the ambiguous case.

Angle Conditions # of Triangles R90��A Aba sin� 0

Aba sin 1

Aba sin! 2 R90!�A ba d 0

ba ! 1 Trigonometric Identities

Pythagorean Identity: 1cossin 22 � TT Quotient Identity: TTT

cossintan

e.g. Prove the identity. TTT 222 cos1cos2sin ��

RS

LS

��

��

��

T

T

TTT

TT

2

2

222

22

cos1cos1

1coscossin1cos2sin

Work with each side separately. Look for the quotient or Pythagorean identities. You may need to factor, simplify or split terms up. When you are done, write a concluding statement.

Since LS=RS then TTT 222 cos1cos2sin �� is true for all values of T .

r y T x

A b c C a B


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

Ch. 6: Sinusoidal Functions Periodic Functions A periodic function has a repeating pattern. The cycle is the smallest complete repeating pattern. The axis of the curve is a horizontal line that is midway between the maximum and minimum values of the graph. The equation is

2min value max value �

y .

The period is the length of the cycle. The amplitude is the magnitude of the vertical distance from the axis of the curve to the maximum or minimum value. The equation is

2min value max value �

a

Trigonometric Functions The graphs of Tsin y and Tcos y are shown below.

Tsin y Tcos y Transformations of Trigonometric Functions Transformations apply to trig functions as they do to any other function. The graphs of cdkay �� )(sin T and cdkay �� )(cos T are transformations of the graphs Tsin y and

Tcos y respectively. The value of a determines the vertical stretch, called the amplitude. It also tells whether the curve is reflected in the T -axis.

The value of k determines the horizontal stretch. The graph is stretched by a factor of k1 . We can use this value

to determine the period of the transformation of Tsin y or Tcos y .

The period of Tky sin or Tky cos is k

R360 , k > 0.

The value of d determines the horizontal translation, known as the phase shift. The value of c determines the vertical translation. cy is the equation of the axis of the curve.

Tsin y Period = 360˚ Amplitude = 1 Zeroes = 0˚, 180˚, 360˚…

Tcos y Period = 360˚ Amplitude = 1 Zeroes = 90˚, 270˚…


2

1.5

1

0.5

-0.5

-1

50 100 150 200 250 300 350 400

g x� � = cos 2�x� �+1

f x� � = cos x� �

1

0.5

-0.5

-1

50 100 150 200 250 300 350 400

g x� � = 0.5�sin x+45� �

f x� � = sin x� �

e.g. e.g.

12cos � Ty � �R45sin21

� Ty

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