Substitution into

Linear Term

Expressions, Brackets

& Substitution

True Eqns

Equations

Multiply Out of

One Linear Bracket

Multiply Out Of

Two Linear Brackets

Solving Eqns by Spotting

Solving Linear Eqns

By Inverses (incl dec)

Solving Quadratics

By Factorising

(Quad Co-eff 1)

Plotting Coordinates

(Negative)

Horizontal & Vertical

Lines

2D Coordinates

(Midpoints & Distances)

Coordinates & Graphs

Factorise Quadratics

(Quad Co-eff 1)

(incl. 0b )

Plotting Linear Graph

(positive coordinates)

Sequences from

Expressions

Forming an

Expression from a

Linear Sequence

Substitution into

Linear Expression

Linear Sequences

(terms & common

difference)

Quadratic Sequences

( 0b )

Solving Linear Eqns

Variables Both Sides

Solving Linear Eqns

Fractional Coefficients

Solving Quadratics

By Factorising

(Quad Co-eff 1 )

Plotting Graph using cmxy

(Parallel Lines)

Sketching Quadratics

by Factorising

(Quad Co-eff 1)

Sequences

Simplifying Quadratic

Algebraic Fractions

&,,

Algebaric Fractions

Factorise Into

One Linear Bracket

Collecting Like Terms

Substitution into

Quadratic Expression

Changing the Subject of

a Formula

Quadratic Sequences

( 0b ) Measuring Gradient )(m

Plotting Quadratics

Graphs

Substitution into

Quadratic Term

Plotting Linear Graph

(negative coordinates)

Solving Eqns by

Trial & Improvement

Solving Simultaneous Eqns

Inequalities

4) Algebra (Variable Numbers)

Formulae

Plotting Coordinates

(Positive)

Factorise Quadratics

(Quad Co-eff 1 )

Completing the Square

Solving Quadratics by

Completing the Square

Solving Quadratics with

Quad. Equations Formula

Regions with Inequalities

Sketching Quadratics (by Completing the Square)

Perpendicular Lines

Sketching Quadratics

by Factorising

(Quad Co-eff 1 )

3D Coordinates

Graph Shape Types

Transformations of

Graphs

Key

Super-Topic

Area

Topic

Topic

Topic

Topic

Steps For Everyone

Genius Steps

Brackets are the 1st, & very

important genius steps in algebra.

Col

lect

ing

Lik

e T

erm

s TOP adding terms

xx 52 x7

2 or more variables

yyxx 3432 yx 75

subtracting terms

yy 59 y4

Sub

stit

utio

n in

to L

inear

Term

TOP x positive Sub 5x into x3

15

53

3

x

x negative

Sub 5x into x3

15

53

3

x

15

53

3

x

Negative Coefficient

Sub 7x

into x3 21

73

3

x

Sub

stit

utio

n in

to Q

uadra

tic

Expr

ess

ion

TOP 2x & constant

Sub 4x

Into 72 x 23

74

7

2

2

x

Quad. Co-efficient not = 1

Sub 7x

into 542 2 xx

121

52898

57472

542

2

2

xx

3 Term Quadratics

Sub 3x

Into 122 xx 16

1323

2

2

2

xx

2x & linear term

Sub 3x

Into xx 22 15

323

2

2

2

xx

Mul

tipl

y O

ut o

f O

ne L

inear

Bra

cket

Constant Multiplier

Multiply

)5(3 x Out of

Brackets

( x 5 )

3 x3 15

153)5(3 xx

Variable Multiplier

Multiply )2( xx

Out of

Brackets

x( )2

x 2x x2

xxxx 2)2( 2

Linear Coefficient not 1 Multiply

)34(2 x Out of

Brackets

x4( )3

2 x8 6

68)34(2 xx

TOP

Factorise

153 x

Into

Brackets

Constant Factor

x( )5

3 x3 15

)5(3153 xx

Fac

tori

se I

nto

One

Lin

ear

Bra

cket

TOP

Factorise

156 x

Into

Brackets

Linear Coefficient Not 1

x2( )5

3 x6 15

)52(3156 xx

Factorise

xx 62

Into

Brackets

Variable Factor

x( )6

x 2x x6

)6(62 xxxx

Expressions, Brackets

& Substitution

Sub

stit

utio

n in

to L

inear

Expr

ess

ion

TOP x positive

Sub 5x

into 43 x 19

453

3

x

Negative Coefficient

Sub 7x into 13 x 20

173

13

x

x negative

Sub 5x into 43 x

11

453

3

x

2 or more Linear Terms

If 5&3 ba

what’s ba 24 2

5234

24

ba

Sub

stit

utio

n in

to Q

uadra

tic

Ter

m

f

TOP

x negative Sub 5x into 2x

25

55)5( 2

2

x

x positive Sub 5x into 2x 25

55)5( 2

2

x

Co-efficient not = 1

Sub 3x into 22x

18

9232

2

2

2

x

Note: 22 )32()2( x

3662

Multiply )3)(2( xx

Out of

Brackets

Positive Constants (in brackets)

x( )2

x( 2x x2

)3 x3 6

6322 xxx 65)3)(2( 2 xxxx

Mul

tipl

y O

ut T

wo

Lin

ear

Bra

cket

TOP

x( )2

x( 2x x2

+ )5 x5 10

10252 xxx 103)5)(2( 2 xxxx

Multiply

)5)(2( xx

Out of

Brackets

Negative Constants (in brackets)

Multiply

)53)(42( xx

Out of

Brackets

Linear coefficient (in brackets) not 1

x2( )4

x3( 26x x12

)5 x10 20

2010226 2 xxx 20326)53)(42( 2 xxxx

x( )5

x( 2x x5

)5 x5 25

25552 xxx 25)5)(5( 2 xxx

Multiply )5)(5( xx

Out of

Brackets

Difference of Squares

)4)(4()4( 2 xxx

x( )4

x( 2x x4

)4 x4 16

16442 xxx 168)4( 22 xxx

Multiply 2)4( x

Out of

Brackets

Complete Squares

Fac

tori

se Q

uadra

tics

(Q

uad C

o-eff

1)

TOP

Factorise

252 x

Difference of Squares

const. = -25 = -5 x 5 = 5 x -5

25

2555

)5)(5(

2

2

x

xxx

xx

so 252 x )5)(5( xx

Factorise

652 xx

All Positive Constants

const. = 6 = 1 x 6 = 6 x 1

67)6)(1( 2 xxxx

6 = 2 x 3 = 3 x 2

65)3)(2( 2 xxxx

so 652 xx

)3)(2( xx

Factorise

542 xx

Negative Constants

const. = -5 = -1 x 5

54)5)(1( 2 xxxx

-5 = 1 x -5

54)5)(1( 2 xxxx

so 542 xx

)5)(1( xx

Sim

plif

ying

Qua

dra

tic

Alg

ebra

ic F

ract

ions

TOP

Simplify

43

3522

2

xx

xx

a not = 1

Factorise:

)43)(1(

)32)(1(

xx

xx

)43)(1(

)32)(1(

xx

xx

43

32

x

x

Simplify

6

232

2

xx

xx

a = 1

Factorise:

)3)(2(

)1(2

xx

xx

)3)(2(

)1(2

xx

xx

3

1

x

x

Fac

tori

se Q

uadra

tics

(Q

uad C

o-eff

no

t =1

)

TOP

Factorise

352 2 xx

a & b both prime

x2 ? 22x x

3 ?

the 22x must be xx2

and 313 or 313

or 133 or 133

372)3)(12( 2 xxxx

352)1)(32( 2 xxxx

372)3)(12( 2 xxxx

352)1)(32( 2 xxxx

)1)(32( xx

Factorise

672 2 xx

a prime, but not c

x2 ? 22x x

6 ?

the 22x must be xx2

and 616 or 166

or 326 or 236

(ignore negs as b & c pos)

6132)6)(12( 2 xxxx

682)1)(62( 2 xxxx

682)3)(22( 2 xxxx

672)2)(32( 2 xxxx

)2)(32( xx

Factorise

9204 2 xx

neither a or c prime

? ? 24x ?

9 ?

the 24x xx 4 or xx 22

and 919 or 199

or 339

(ignore negs as b & c pos) Trying all combos gives:

9204)92)(12( 2 xxxx

)92)(12( xx

Com

plet

ing

the

Squ

are

b even (a = 1)

complete

the square 762 xx

b = 6, so square x + 3 96)3( 22 xxx

7616)3( 22 xxx

16)3( 2 x

b odd (a = 1)

complete

the square 252 xx

b = 5, so square x +2.5 25.65)5.2( 22 xxx

2525.4)5.2( 22 xxx

25.4)5.2( 2 x

a not 1

complete

the square

9122 2 xx

)6(2122 22 xxxx

18122)3(2 22 xxx

91229)3(2 22 xxx

9)3(2 2 x

TOP

+, -

, x, ÷

Alg

ebra

ic F

ract

ions

TOP

+ & - Algebraic Fractions

Simplify

2)1(

1

1

xx

x

2)1(

1

1

xx

x

Make denom. the same

22 )1(

1

)1(

)1(

xx

xx

22

2

)1(

1

)1(

xx

xx

2

2

)1(

1

x

xx

Flip 2nd fraction & x

2

)3(2

)1(2

2

x

x

x

x

)1)(2(2

)3)(2(2

xx

xx

)1)(2(2

)3)(2(2

xx

xx

1

3

x

x

x & ÷ Algebraic Fractions

Simplify

)3(2

2

)1(2

2

x

x

x

x

Solve 11254 xx

x s Both Sides

3

62

51124

11254

x

x

xx

xx

Sol

ving

Lin

ear

Eqn s

Var

iable

s B

oth S

ides

TOP

Solve

12743 xx

Biggest Coefficient on RHS

x

xx

xx

416

37124

12743

So 4x

When 4x ,

which of these

eqns is true?

1) 205 x

2) 93 x

3) 287 x

4) 112 x

Linear Term Equals Constant

1)

2020

2045

205x

True

2)

915

953

93x

False

3)

2828

2847

287x

True

4)

118

1142

112x

False

1) 205 x &3) 287 x

Are both true.

Tru

e E

qn s

TOP

Variable & Constant Equals Constant

When 5x ,

which of these

eqns is true?

1) 207 x

2) 122 x

3) 41x

4) 49 x

1)

2012

2075

207xFalse

2)

123

1225

122x

False

3)

44

415

41x

True

4)

44

495

49x

True

3) 41x

& 4) 49 x

Are both true.

When 2x ,

which of these

eqns is true?

1) 735 x

2) 1214 x

3) 2057 x

4) 1153 x

Linear Expression Equals Constant

1)

77

7325

735x

True

2)

129

12124

1214x

False

3)

2019

20527

2057x

False

4)

1111

11523

1153x

True

1) 735 x

& 4) 1153 x

Are both true.

1243

123

x

so 4x

Solve

123 x

Linear Term = Constant

Sol

ving

Equ

atio

ns b

y S

pott

ing

Sol

utio

ns

TOP

Solve

152 x

Linear Expression =

Constant

1532

152

x

So 3x

Solve

65 x

Variable & Constant

= Constant

6511

65

x

so 11x

25555

25

2

2

x

25)5()5()5( 2

so 5x or -5

Solve

252 x

Quadratic Term =

Constant

Sol

ving

Lin

ear

Eqn s

By

Inv

ers

es

(inc

l dec)

TOP

so 5.2x

or 5.2x

25.6

252

x

x

Solve

25.62 x

Quadratic Term = Constant

)&( 2x

Solve 22.263 x

Linear Term = Constant )&(

So 74.8x 3

22.26

22.263

x

x

3 3

Solve 4.697.5 x

Variable & Constant = Constant

(+ & -)

so 37.12x

97.54.6

4.697.5

x

x

+5.9

7 +5

.97

Solve 4.1252 x

Linear Expression = Constant

(Two-Step)

So 7.8x

2

4.17

4.172

54.122

4.1252

x

x

x

x

5 5

2 2

For

mulae

TOP

Sub x = 3

into 25 xy

Linear, Single Variable

215

235

25

y

y

xy

13y

Sub x = 3,

a = 5, b = 7 into

abxy 22

Multi-Variable

709

7523

2

2

2

y

y

abxy

97y

Sub y = 55 ,

a = 3, b = 7 into

axby 2

to find x

Substitution Forming an

Equation to Solve

x

x

x

axby

36

34955

3755 2

2

2x

Equations

Make a

the

subject of baF 2

2 or More Steps

bFa

baF

2

2

2

bFa

Chan

ging

the S

ubje

ct o

f a

For

mul

a

TOP

Find the

inverse of 23 xy

Finding the Inverse

3

2

3

2

32

23

xy

xy

xy

xy

)7(2&72

3

2&23

xyx

y

xyxy

Make a the

subject of

baF 22

With Quadratics

2

2

2

2

2

2

bFa

bFa

baF

2

bFa

Make a the

subject of maF

One Step

maF

m

Fa

Sol

ving

Ine

qual

itie

s

TOP

Represent

2x &

4x

on a

number

line

Representing on a No Line

4x

-4

2x

2

Solve the

inequality

1373 x

Solving Inequalities

Treat Like Equations

So 2x

2

63

1373

x

x

x

7 7

3 3

4

4

123

x

x

x

X by -1

Flip

Ineq.

Solve the

inequality

123 x

Sign Changes

So 4x

Tri

al &

Im

rpvo

em

ent

TOP

Solve

263 23 xx

To 1dp using

trial of

improvement

Trial and Improvement

578.2625.2

037.283.2

168.252.2

491.221.2

375.345.2

543

202

263 23

x

x

x

x

x

x

x

xx

So x = 2.2 (1dp)

So 2 < x < 3

So 2.2 < x < 2.3

So 2.2 < x < 2.25

Solve

832

x

Unit Fractions

10

52

832

x

x

x

Sol

ving

Eqn

s F

ract

iona

l C

oeff

icie

nts

TOP

Solve

1124

3x

Unit Fractions

12

363

94

3

1124

3

x

x

x

x

Sol

ving

Qua

dra

tics

by

Fac

tori

sing

(Q

uad C

o-ef

f 1)

TOP

Solve xx 5282

Needs Rearranging First

xx 5282

so 0652 xx

& 0)3)(2( xx

Hence either:

Hence 2x or 3x

02 x

so

2x

03 x

so

3x

Solve 0832 xx

Factorise 1st (Quadratic = 0) Factorise L.H.S. of

01032 xx 0)2)(5( xx

Hence either:

Hence 5x or 2x

05 x

so

5x

02 x

so

2x

Sol

ving

Qua

dra

tics

Usi

ng t

he

For

mul

a

TOP The Formula

State the

quadratic

equations

formula

if 02 cbxax

a

acbbx

2

42

Integer Solutions

Solve

062 xx

using the

formula

12

)6(1411 2

x

2

251x

so 2

251x

so 2x or 3x

Decimal Solutions

Solve

0432 2 xx using the

formula

22

)4(2433 2

x

4

413x

4

403.63x

so 403.9x

or 403.3x

Sol

ving

Qua

d b

y Com

plet

ing

the

Squ

are TOP a = 1

solve 0762 xx

by

completing

the square

0762 xx

16)3(

016)3(

2

2

x

x

43 x or 43 x 1x or 7x

a not 1

solve 0342 2 xx

by

completing

the square

0342 2 xx

05)1(2 2 x

5.2)1(

5)1(2

2

2

x

x

so 581.11x or 581.11 x

581.0x or 581.2x

Sol

ving

by

Fac

tori

sing

(a

not

1)

TOP

Solve 0372 2 xx

a 1 Factorise L.H.S. of

0372 2 xx 0)3)(12( xx

Hence either:

Hence 5x or 2x

2

112 x

so

2

1x

03 x so

3x

Sol

ving

Sim

ulta

neou

s E

quat

ions

TOP

1975 yx

1135 yx

84 y

2y

sub y = 2 into

19145 x

55 x

1x

1x & 2y

Solve

these

simul-

taneous

equations

1135

1975

yx

yx

Cancelling by subtraction

b

- - - a

a-b

a

a when y =23

2432 yx

933 yx 155 x

3x sub x = 3 into

2436 y

183 y

6y

3x & 6y

Solve

these

simul-

taneous

equations

933

1432

yx

yx

Cancelling by addition

b

+ + + a

a+b

a

a when x = 3

2725 yx

1734 yx

81615 yx

3468 yx

11523 x 5x

sub x = 5 into 27225 y

22 y

1y

5x & 1y

Solve

these

simul-

taneous

equations

1734

2725

yx

yx

Change terms to be cancellable

b

+ + +

a

a when x = 5

2b

3a

3a+2b

a

Plot

3x

&

7y

Horizontal & Vertical Lines

Hor

izon

tal &

Vert

ical

Lin

es

TOP

Plot

ting

Nega

tive

Coo

rdin

ates

TOP

Plot

Coord

(-5,-3)

Negative Coordinates

x

y

Plot

ting

Lin

ear

Gra

ph (

nega

tive

coo

rdin

ates)

TOP Finding the Coordinates (from a table)

12 xy x 0 1 2 3 4

Calc. 2x0 +1 2x1 +1 2x2 +1 2x3 +1 2x4 +1 y 1 3 5 7 9

Coords (0,1) (1,3) (2,5) (3,7) (4,9)

Find

some

coords

for the

line 12 xy

Plotting the Line

Plot

the line 12 xy

(using

above

table)

Coordinates & Graphs

Plot

ting

Pos

itiv

e C

oord

inat

es

TOP

Plot

Coor

d

(5,3)

Positive Coordinates

x

y

(the stairs) (the corridor)

Find the

gradient

of this

graph

through

(0,1) &

(5,11)

Measuring Gradient m (Steepness)

Mea

suri

ng G

radie

nt

TOP

Find the

gradient

of the line

between

points

(0,1)

&

(5,11)

For two points 2211 ,&, yxyx

the gradient, 12

12

xx

yym

(change in y , over change in x )

05

111

m

So 2m

Gradient )(m Using the Formula

2D

Coo

rdin

ates

(Mid

poin

ts &

Dis

tanc

es)

TOP

Find the

distance

between

)2,1(

& )6,4(

Distance (between two points)

Right Angled Triangle

so use Pythagoras

5

25

169

43

)26()14(

22

22

Find the

midpoint

of 6,5&8,3

Midpoint (of two points)

7,4

2

14,

2

8

2

68,

2

53

Average

of x-coords Average

of y-coords

Plot

ting

Lin

ear

Gra

ph (

nega

tive

coo

rdin

ates)

TOP

Finding the Coordinates (from a table)

12 xy x -5 -2 0 2 4

Calc. 2x-5 +1 2x-2 +1 2x0 +1 2x2 +1 2x4 +1

y -9 -3 1 5 9

Coords (-5,-9) (-2,-3) (0,1) (2,5) (4,9)

Find

some

coords

for the

line 12 xy

Negative Linear Coefficient (Gradient)

12 xy x -4 -2 0 2 5

Calc. -2x-4 +1 -2x-2 +1 -2x0 +1 -2x2 +1 -2x5 +1

y 9 5 1 -3 -9

Coords (-4,9) (-2,5) (0,1) (2,-3) (5,-9)

Plot

the line 12 xy

Plotting the Line

Plot

the line 12 xy

(using

above

table)

Oops! a mistake

(Not Straight)

Plot

ting

Qua

dra

tic

Gra

phs

TOP Finding the Coordinates (from a table)

322 xxy x -2 -1 0 1 2 3 4

(-2)2

-2x(-2)

-3

(-1)2

-2x(-1)

-3

(0)2

-2x0

-3

12

-2x1

-3

22

-2x2

-3

32

-2x3

-3

42

-2x4

-3

4 +4 -3 1 +2 -3 0 -0 -3 1 -2 -3 4 -4 -3 9 -6 -3 16 -8 -

3

y 5 0 -3 -4 -3 0 5

Coords (-2,5) (-1,0) (0,-3) (1,-4) (2,-3) (3,0) (4,5)

Fill in the

table for 322 xxy

(& hence

find some

coordinates

to plot)

−3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

x

y

Plotting the Line

Plot

the line

322 xxy

(using

above

table)

x-coefficent

positive means

smiley line

Negative Quadratic Coefficient

29 xy x -3 -2 -1 0 1 2 3

9

-(-3)2

9

-(-2)2

9

-(-1)2

9

-(0)2

9

-12

9

-22

9

-32

9 - 9 9 - 4 9 - 1 9 - 0 9 - 1 9 - 4 9 - 9

y 0 5 8 9 8 5 0

Coords (-3,0) (-2,5) (-1,8) (0,9) (1,8) (2,5) (3,0)

Plot

the line 29 xy

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

x

y

x-coefficent

negative means

unhappy line

Given 12 xy

(dashed)

sketch 12 xy

Negative Gradients ( 0m ) & Steepness y = 2x + 1 where m = 2 and c = 1

y = -2x + 1 where m = -2 and c = 1

y -intercepts are both 1 so (0, 1) Gradients (m) are negatives of each other

hence same slope but downhill

Given 12 xy

(dashed)

sketch 32 xy

Parallel Lines & c is y-intercept

y = 2x + 1 where m = 2 and c = 1

y = 2x -3 where m = 2 and c = -3

m the same (2) for both so parallel.

y-intercepts are c, so (0, 1) and (0,-3)

Plot

ting

Gra

ph u

sing

y =

mx

+ c

(P

aral

lel Lin

es)

Given 12 xy

(dashed)

sketch 13 xy

and

12

1 xy

Different Gradients (m) & Steepness

y = 2x + 1 where m = 2 and c = 1

y = ½x + 1 where m = ½ and c = 1

y = 3x + 1 where m = 3 and c = 1

y -intercepts are all 1 so (0, 1)

m = 3 is steepest, m = ½ is least steep

TOP

Ske

tchin

g Q

uadra

tics

by

Fac

tori

sing

(Q

uad C

o-eff

1)

y intercept (c)

Sketch the graph

y = x2 +2x – 8

& label the y intercept

y = x2 + 2x – 8

The y axis is the line x=0

y = 02 + 2 x 0 – 8 = – 8

- -8

x intercepts (roots)

Sketch the graph

y = x2 +2x – 8

& label the x intercepts

y = x2 + 2x – 8

The x axis is the line y=0

0 = x2 + 2x – 8

0 = (x + 4)(x – 2)

so x + 4 = 0 or x – 2 = 0

so x = -4 or x = 2

- -8 2 -4

TOP

Regi

ons

wit

h I

nequ

alit

ies

TOP

horizontal & vertical Lines

Shade out the

unwanted regions to leave regions

2x and

1y

2x

1y

diagonal lines

Shade out the

unwanted region to

leave region

2 xy

2 xy

or solid line

< or > dashed line

partially & completely enclosed regions

Shade out the

unwanted regions to leave regions

24 x and

1&

,3

,1

y

xy

x

24 x

1

3

1

y

xy

x

Ske

tchin

g Q

uadra

tics

by

Fac

tori

sing

(a

not

1)

TOP y intercept (c)

Sketch the graph

y = 2x2 + x – 3

& label the y intercept

y = 2x2 + x – 3

The y axis is the line x=0

y = 2x02 + 0 – 3 = –3

- -3

x intercepts (roots)

Sketch the graph

y = 2x2 + x –3

& label the x intercepts

y = 2x2 + x – 3

The x axis is the line y=0

0 = 2x2 + x – 3

0 = (2x +3)(x – 1)

so 2x + 3 = 0 or x – 1 = 0

so x = -1.5 or x = 1

- -3 1 -1.5

Gra

ph S

hap

e T

ypes

TOP Quadratics, Cubics & Reciprocals

Write

the

equations

of these

graphs on

the

correct

graph on

the right. y=3x

2

y=-2x2

y=x3

y=-5x3

y=7/x

& y=-3/x

23xy

35xy 3xy

xy

7

22xy

xy

3

Perp

end

icul

ar L

ines

TOP Finding Gradients A line has

gradient 3,

what is the

gradient of a

line

perpendicular

to this?

Perpendicular Lines

have gradients which

multiply to make -1

(negative reciprocals)

13

13

Gradient = 3

1

Finding Gradients

Find the eqn

of the red

line with

y-intercept

(0,-1)

perpendicular

to y = -2x + 4

12

12

so gradient is ½

y = ½x – 1

y=-2x+4

y=?

3D

Cor

rdin

ates

TOP

52534 22 AC

AC = 5

Using the Coords to Find a Distance

On cuboid ABCDEFGH Find the

length of

diagonal

AC? A(0,0,0) B(4,0,0)

C(4,3,0) D

E F

G(4,3,2) H

x

y

z

4

3

A B

C

AB = 4 - 0 = 4

BC = 3 – 0 = 3

D(0,3,0) E(0,0,2) F(4,0,2) H(0,3,2)

(x, y, z) coordinates

Cuboid ABCDEFGH has points

A,B,C & G

labelled.

What are

D,E,F&H? A(0,0,0) B(4,0,0)

C(4,3,0) D

E F

G(4,3,2) H

x

y z

Ske

tchin

g Q

uadra

tics

by

Com

plet

ing

the

Squ

are

TOP Intercepts (by solving y = 0)

Find the

axis intercepts

of y=x

2+4x-5

y-intercept is (0,-5)

complete square and solve 44)2( 22 xxx

549)2( 22 xxx

now solve 09)2( 2 x 9)2( 2 x

32 x or 32 x

so 1x or 5x

(0,-5) (1,0) & (-5,0)

9)2(54 22 xxx

Min is when 0)2( x

so Min at x = - 2

so min at (-2, -9)

Max and Min

Find the

minimum

point of y=x

2+4x-5

and

sketch

the graph

(1,0) (-5,0)

Min=(-2,-9) (0,-5)

Tra

nsfo

rmat

ions

of

Gra

phs

TOP

-f(x) reflects in x-axis

f(-x) reflects in y-axis

Negative multiples of x or y

The black

line is y=f(x), show

y=-f(x) in

red and y=f(-x)

in green

y=f(x)

y=-f(x)

y=f(-x)

(-3,1)

(3,-1)

(3,1)

x

y

f(x-3) moves 3 right f(2x) stretches x½ in x-direction

x

y

x - direction

The black

line is y=f(x), show

y=f(x-3) in

red & y=f(2x)

in green

y=f(x) y=f(x-3)

y=f(2x)

(0,7) (0,4) (0,2)

f(x)-3 moves down 3

2f(x) stretches x2 in y-direction

x

y

y - direction

The black

line is y=f(x), show

y=2f(x) in

red and

y=f(x)-3 in

green

y=f(x)

y=2f(x)

y=f(x)-3

(0,-2)

(0,1)

(0,2)

Express

3n+1

as a

sequence

Linear Coefficient Positive n 1 2 3 4 5

3n+1 3x1+1 3x2+1 3x3+1 3x4+1 3x5+1

4 7 10 13 16

Sequence is 4, 7, 10, 13, 16…

Sequ

enc

es

from

Expr

ess

ions

TOP

Find the

7th & 20th

terms of

3n+1

Finding a particular term

for 7th term n = 7

7th term = 3x7 + 1 = 22

for 20th term n = 20

20th term =3x20 + 1 = 61 7th term = 22, 20th term = 61

Express

-3n+7

as a

sequence

Linear Coefficient Negative

n 1 2 3 4 5

-3n+7 -3x1

+7

-3x2

+7

-3x3

+7

-3x4

+7

-3x5

+7

4 1 -2 -5 -8

Sequence is 4, 1, -2, -5, -8…

Find the common

difference for:

2n + 1, 3n + 1

4n + 1, 5n + 1 What do you notice?

Common Difference & Linear Coefficient 2n + 1 = 3, 5, 7, 9, 11… Com Dif = 2

3n + 1 = 4, 7, 10, 13, 16… Com Dif = 3

4n + 1 = 5, 9, 13, 17, 21… Com Dif = 4

5n + 1 = 6, 11, 16, 21, 26… Com Dif = 5

Common difference is always

the linear coefficient

For

min

g an

Expr

essi

on f

rom

a L

inea

r S

equ

enc

e

TOP

Form an

expression for

sequence

5, 7, 9, 11, 13…

Expressing linear sequences

5, 7, 9, 11, 13

Common difference is 2

so sequence is 2n + ?

1st term of 2n = 2x1 = 2 we need to + 3 to make our 1st term of 5

5, 7, 9, 11, 13… = 2n + 3

+2 +2 +2 +2

Find the 100th

term of

5, 7, 9, 11, 13…

Finding a particular term

5, 7, 9, 11, 13

Common difference is 2

so sequence is 2n + ?

1st term of 2n = 2x1 = 2 we need to + 3 to make our 1st term of 5

5, 7, 9, 11, 13… = 2n + 3

100th term when n = 100

100th term = 2x100 + 3

100th term = 203

+2 +2 +2 +2

Lin

ear

Sequ

enc

es

(term

s &

com

mon

dif

fere

nce)

TOP

Find the next

3 terms of

sequence: 20, 18, 16, 14, 12…

Negative Common Difference

You just keep subtracting 2

20, 18, 16, 14, 12, 10

next three terms are:

10, 8, 6

-2 -2 -2 -2 -2

Find the

next 3

terms of

sequence: 3, 5, 7, 9, 11…

Common Difference

You just keep adding 2

3, 5, 7, 9, 11, 13

next three terms are:

13, 15, 17…

+2 +2 +2 +2 +2

3, 5, 7, 9, 11…

1st term = 3, 4th term = 9

Find the 1st

& 4th terms

of sequence:

3, 5, 7, 9, 11…

Terms as Position

5th term

1st term

2nd term

3rd

term

4th term

Sequences

Qua

dra

tic

Sequ

enc

es

(b =

0)

TOP

Express the

nth term of 1,4,9,16,25...

and

6,12,27,48,75...

ax2

1, 4, 9, 16, 25

+3 +5 +7 +9

+2 +2 +2

3, 12, 27, 48, 75

+6 +6 +6

+9 +15 +21 +27

Second differences

are constant 6.

Try 2

6 n2 = 3n2

3n2=6,12,27,48,75...

6,12,27,48,75...= 3n2

Second differences

are constant 2.

Try 2

2 n2 = n2

n2 = 1,4,9,16,25...

1,4,9,16,25...= n2

Express the

nth term of 5,11,21,35,53...

ax2 + c

5,11,21,35,53... = 2n2 + 3

Try 2

4 n2 = 2n2

2n2 = 2,8,18,32,50...

Need to add 3 to

each term to give

5,11,21,35,53...

so it’s 2n2 + 3

5, 11, 21, 35, 53

+4 +4 +4

+6 +10 +14 +18

Second differences

are constant 4.

Qua

dra

tic

Sequ

enc

es

(b n

ot=

0)

TOP

Express the nth

term of

4,13,26,43,64...

Complex Quadratics

Second differences

are constant 4.

Try 2

4 n2 = 2n2

2n2 = 2, 8, 18, 32, 50...

Seq = 4, 13, 26, 43, 64...

Diff = 2, 5, 8, 11, 14

Goes up by constant 3,

try 3n = 3, 6, 9, 12, 15

Need to -1 to make Diff.

2, 5, 8, 11, 14 = 3n-1

4,13,26,43,64...

= 2n2 + 3n – 1

+3 +3 +3 +3

4, 13, 26, 43, 64

+4 +4 +4

+9 +13 +17 +21

+2 +14