[PDF] 2 How to use this booklet

2 How to use this booklet

1 2 How to use this booklet This booklet is designed to clarify the content prescribed for Mathematics In addition, it also provides some tips on how you should tackle real problems on a daily basis

AS/A Level Mathematics Functions - Maths Genie

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1

2. How to use this booklet

This booklet is designed to clarify the content prescribed for Mathematics. In addition, it also provides

some tips on how you should tackle real problems on a daily basis. Candidates will be expected to have

mastered the content outlined for grades 8-11. This booklet must be used to master some mathematical

rules you may not aware of. The prescribed textbook must also be used.

3. Study and examination tips

• Generalize, make conjectures and try to justify or prove them. • Develop problem-solving and cognitive skills. • Make use of the language of Mathematics. • Identify, investigate and solve problems creatively and critically.

• Use the properties of shapes and objects to identify, investigate and solve problems creatively and

critically.

• Encourage appropriate communication by using descriptions in words, graphs, symbols, tables and

diagrams. • Practice Mathematics every day.

4. OVERVIEW OF FUNCTIONS

Functions

*HQHUDOGH¿QLWLRQ

Types of functions

Linear

function y = ax + qy = a(x + p) 2 + qy = ab x+p + q

Quadratic

function

Inverse

function

HyperbolaExponential

a y = + q x + p 2

Properties of functions:

Axis of symmetry

Domain

Range

Notation

y = ax + q y = a(x + p) 2 + q y = ab x+p + q a y = + q x + p a > 0 a > 0

5.1 STRAIGHT LINE

General representation or equation

y = ax + q or y = mx + x. a or m is the gradient and q or c is the y - intercept a < 0 a = 0 a > 0 aLVXQGH¿QHG q < 0 y = q q < 0 there is no q-value Domain and range is xúR and yúR respectively. y y y y xx x x 3

5.2 HYPERBOLA

General representation or equation

Dotted lines are asymptotes Dotted lines are asymptotes • q is the vertical translation • p is the horizontal translation • For y = a x , p = 0 and q = 0. The vertical asymptote is and the horizontal asymptote is y = 0. The axes of symmetry are y = x (Positive) and y = -x (Negative). • For y = a x + q, p = 7KHYHUWLFDODV\PSWRWHLV[ DQGWKHKRUL]RQWDODV\PSWRWHLVy = q. The axes of symmetry are y = x + q (Positive) and y = x + q (Negative). • For y = a x + q, p = 0 the vertical asymptote is x =DQGWKHKRUL]RQWDODV\PSWRWHLVy = q. The axis of symmetry is . • For y = a x ST¹(y - q) (x + p) = a the vertical asymptote is x + - p and the horizontal asymptote is y = q. The axis of symmetry is y = ± (x + p) + q.

Example 1

3

Given f (x) = + 1

x - 2

1.1 Write down the equations of the asymptotes of f.

1.4 Determine the domain and the range of f.

1.5 Determine the decreasing and increasing functions of the axes of symmetry of f.

1.6 Draw a sketch graph of f.

a a + q a + q y = a + q y = x or y = x or y = x - p or x + p 4

6ROXWLRQ

1.1 y = 1

x = 2 1.2 3 y = + 1 x - 2 3 = + 1 x - 2 - x + 2 = 3 x = - 1 1.3 3 y = + 1 1 y = - 2 ,QFUHDVLQJD[LVRIV\PPHWU\y = x + c

1 = 2 + c

c = -1 y = x -1 'HFUHDVLQJD[LVRIV\PPHWU\y = x + c

1 = -2 + c

y = - x + 3 1.6 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 y x 14 12 10 8 6 4 2 -2 -4 -6 -8 -10 5

Activity 2.1

*LYHQ h(x)= 12 + 6 for x > x - 4

1.1 Draw a neat sketch graph of h in your workbook. Show all intercepts with the axes and asymptotes.

1.2 Write down the equation of k if kLVDUHÀHFWLRQRIKDERXWWKHx-axis.

2. The diagram above shows the graph for f (x) =

a + q x + p The lines x = -1 and y = 1 are the asymptotes of f3LVDSRLQWRQIDQG7LVWKHx-intercept of f.

2.1 Determine the values of a, p, and q.

&DOFXODWHWKHFRRUGLQDWHVRI7WKHx-intercept of f.

2.3 Determine the value of c if the graph of f is symmetrical with respect to the line .

5.3 PARABOLA

1RWH • You need to know how to solve quadratic equations, in order to be able to deal with parabola. • The standard form of a quadratic equation is ax 2 +bx + c = • If a quadratic equation is already factorized and one side is equal to zero, then the equation is in standard form and you should just write the answers from the given factors. question in standard form and then factorize. • When the question says correct to one or two decimal places, you are expected to solve the quadratic equation using the quadratic formula. • Then use the graphic (draw the sketch of the parabola) or number line method. ([DPSOHV

1 Solve for x:

x(x-1) =

6ROXWLRQ

x = or x-1= x = or x= 1 x y -1T 3 1 6

2 Solve for x

(x-3)(x+5) = 9 x 2 + 5x - 3x - 9 = x 2 + 2x - x =

E“¥E

2 -4ac 2a x = - 2“¥4 2 -4(1)(-9) 2(1) x = 2,16 or x = -5,16

3 Solve for x

15x - 4 > 9x

2

15x - 4 - 9x

2 9x 2 - 15x + 4 (3x - 1) (3x

1 4

< x <

3 3

OOy{ 1 3 4 3 7

Activity 3.1

1. Given x

2 + 2x =

1.1 Solve for x

1.2 So, you must determine the values of x for which x

2 •x

2. Solve for x:

2x 2 - 3x - FRUUHFWWR7:2GHFLPDOSODFHV

3. Solve for x

¥[- 2 + x = 4

ANSWERS

1.1 x RUx = -2

1.2 x"RUx•

2. x = 2,77 or x = -1,27

3. x = 3 is the only solution

3.2 General representation or equation

y = ax 2 or y = ax 2 + q or y = a (x + p) 2 + q or y = ax 2 + bx + c

Important deductions

for a < 0 for a > 0 • For y = ax 2 , p = 0 and q = 0, the turning pointLVDQGy-intercept is y = 0 The domain is xúR and the rangeLV\•yúRLID!RU\"yúR or R if a < 0 • For y = ax 2 + q, p = 0, the turning pointLVTDQGy-intercept is y = q The domain is xúR and the rangeLV\•TyúRLID!RU\"yúR if a < 0 • For y = a (x + p) 2 + q, the turning pointLVSTDQGy-intercept is y = a (p) 2 + q The domain is xúR and the rangeLV\•TyúRLID!RU\"yúR if a < 0 • For y = ax 2 + bx + c, the turning point is - b 4ac - b 2 2a 4a and y-intercept is y = c • The domain is xúR and the range is

4ac - b

2 ; yúR if a > 0 or 4a

4ac - b

2 ; yúR if a < 0 4a The roots or x-intercepts are determined by equating y to zero and solving for x. 8

3.3 Sketch the graph of a parabola: You need:

• the y-intercept (here x • the x-intercepts (here y • the axis of symmetry (here x = - b obtained by noting that at the turning point of 2a y = ax 2 + bx + c = ZHKDYH dy , so 2ax + b = . That means x = b dx 2a. • the maximum/ minimum value is obtained by substituting the axis of symmetry in the given equation. ([DPSOH

Sketch the graph of y = f(x) = x

2 - 5x - 6 y-intercept

If x = WKHQ f = -6

x-intercepts

If y = WKHQ x

2 - 5x (x - 6) (x x = 6 or x = - 1

The axis of symmetry

x = -b , so x = - 5

2a 2(1)

5 x = - 2

Corresponding y-value (maximum/ minimum)

f 5 5 2 5 - 6

2 2 2

= 12 1 4 )LQGLQJWKHHTXDWLRQRIDJLYHQSDUDEROD • given the roots x 1 and x 2 , use y = a (x - x 1 )(x - x 2

‡JLYHQWKHWXUQLQJSRLQWSTXVHy =a(x - p)

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