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Year 10 Math Summary

PRODUCTS AND FACTORS

QUADRATIC FORMULA

• b = x4ac

y =^ 2x"^ +^ 3x^ +^ 6

1 N

(^9) b Examples

1 2x2^ +^ 4x^ +^2 =^0

• 4 1 42 +^ 4(2)(2)

2(2)

= O^ => - O

• 20 . = = + = -

L L

x

= - 1

COMPLETING THE (^) SQUARE x2 (^) + (^) 8x -^20 = 0 x + (^) 8x = 20 L move's^ to the other side

x2 + 8x + 16 =^20 +^164 half'b' the^ Square the

(x + 4) =^36 factorise^ and^ number

x + 4 = 26 take (^) the square root^ of both

x +^4 =^ -^6 Sides

x =^ -^16

Solve (^) for (^) negative and^ positive x +^4 =^6

x =^2

x =^ 20r^ -^10

NON MONIC

4x +^ 4x^ - 3 =^0

x + x - q =^0 Divide all the terms by 'al

x + x =^ - move's^ to^ the^ right side (^) of the (^) equation x + x +^ + =^1 (x + (^) z) = IT complete^ the

Square as^ usual

x =^ zor -^22

LINEAR RELATIONSHIPS

Identifying linear^ candy-intercepts

Example

3) - 2y =^5

When x =^0

y = - When (^) y = x (^) =

Midpoint formula

-his is^ the^ halfway point between^ two^ points

M = (^) x (^) , + (^) Cy , y^ ,^ +^ yz

Example

(18 , 10)^ (10 , - 4)

18 +^1016 I^ -^4 2 2 = (^14) , 6

Distance Formula

d (((2 -^ x , )+ (yz - y ,)

Examples

I 2

0. -3 5 , 2

5 - 03 +^ (2 + 3)

(5) +^ (5)

Determine whether^ the^ triangle by^ joining Points A^ (-1^ ,^ 7)^ BC-1^ ,^ -3)^ and^ JC-7 , 01)^ is^ Scalene^ , isosceles^ or

equila +^ eral^ -

AB d = 10 units vertical interval

Bcd =^ F+^ +^ (01^ +^ 3)

= (- 6)2 + (2)

I 40 units

A(a (^) = ( -^ z + 1) +^ ( -^1 - 7)

= ( - 6) + (- 8)

100 units

• 10

AB = Ac^ =^10 units ... ABC is isosceles

Gradient formula

Gradient is defined as^ the^ ratio^ , the^ vertical^ rise^ to^ horizontal^ run^ between^ any^2 points^ on^ a^ line^.

Gradient =^ m^ = rise = yz - 3 ,

run (^) x2 -^ x Example

(-10 , - 8) (6, 8)

8 + (^8)

6 + 10 =

I (^) I (^) gradient y =^ mx^ +^ C

Y intercept

Point gradient formula

y - y ,^ =^ m(x^ -^ x)

x, and y ,^ can^ be^ any^ point

Examples (2 (^) ,^ -^ 8)m^ = t

y +^8 =^ - 5(x - 2

• 5y - 40 = C - 2

c + 5y + 38 = 0 /general form)

y = - - 38 (form^ y = mx + (^) )

(7, 3)^ (10^ ,^ 6)

=

y -^6 =^ 1(x - 10)

y =^ c^ -^4 (formy^ =^ mx^ +

x - y -^4 =^ 0) general form /

Parallel and Perpendicular lines

If two lines^ with^ gradients my and me are parallel then my and my are equal

m (^) , = (^) mz

if two lines with^ gradient m^ , and^ me are perpendicular then^ when^ multiplied^ would^ equal-1' or negati

• (^) ve recriprocal

m , X mz =^ -

mi (^) =

INDICES AND SURDS

Inde law (^) summary a xam (^) = an +^ m

2(an)m = anxm/nm

3ah = a^ = am^ -^ n

L a = 1

5 a^ -^ m^ =^ am

6 amm^ =^ ma

U

7 am =^ M^ A

Examples

4) mon7 xmnx5mn2^ = 20m8n

2 ↓

• 8 yea - 2 y 3 "456p

81r

-x0 - 4(2y)0 = 3

5 , 3 = 7x -^3

• x^ -^3 I

= -x

63x - 2 = (3x -^ 2)t

I I

7 314994414 = 144-115 =^ 342-19-

(p +^ b) pa (^) + + (^) +

p2 + 2p^ -^ z^ +^ p -

Rational and irrational^ numbers A (^) number is rational when it can be written (^) as (^) integers

5 ,, ,, 0. 17

A number is irrational When take (^) the (^) square roof of a non perfect (^) square

Simplifying Surds

Example

g 22

= 3

= 3 Multiplying and^ dividing surds

Examples

• 2822 - 320
  • 416 + 6160

= -^16 +^2410

(^82) -^4 226 13 Adding and^ Subtracting^ Surds

Examples

12 45 - 48 - 5

= 23 -^35 -^42.^5

= - 23 - 45

48 +^3147 +^ 5 M

• 163 + 213 + (^103)

= 473

Rationalising the^ denominator

Example

5 -^2 - 2 + (^5) 3 -^2 3 +^1

15 -^6 +^10 -^2 23 +^3 -^2 -^3

I 2 215 -^26 +^210 -^4 -^23 - 3 -^1 2 = 215 - 26 +^210 - 3 -^5 2

Surds in^ alegbraic terms

Examples

2 +^3 5 +^33

= 10 +^36 +^35 +^93

2x +^ y(x - 3y

2x - 6xy + xy - by

= (^) 2x - 5x(y - 3y

FINANCIAL MATH

Simple interest

-interest

rate

Formula T^ = Pwn >^ no^ of period

interest prinicple

Examples

I Simple^ interest^ if^ $^50 ,^000 is^ invested^ a t^6 %^ pa^.

I = Prn = (^) 50000X X

=> 59000

ii How^ much^ money will^ be^ recieved at^ the^ end^ of an investment

50000 +^9000

= 559000

2 Nick^ receives^ $^867 in^ total^ interest^ after^ investing^ his^ money^ at^1.^5 %^ for^2 years^.^ How^ much^ is^ invested^?

I =^ Prn^ I^ =^867 r^ =^1.^5 %^ P^ =^2

867 =^ Px x

p = 867 + 0.^03 P (^) = (^) $ 28900

Compound interest

Formula A = P If^ n no^ of period in + erest

amount^ Total^ Prinpicle

I interest Ov present value

Fu + uve^ T^ =^ P1^ +^ r)

value Examples a Blake^ invests^ S7000^ over^5 years at^ a^ compound interest rate^ of 4.^5 % P. a·^ Calulate^ the^ f after (^5) years

fu =^ Pu^1 +^ R)h

fu =^ 7000(1^ +^ 45/

fu =^ S8723^.^27

↓Calculate the^ pu of an^ annuity that^ has^ fu of $ 500 , 000 over (^8) years with^ interest^ rate^ of^8.^5 %^ p^.^ a^ com

• pounded (^) monthly

r = 0.^085 = 12 fu =^ Pv (1 + v)h

n = 8x12^ =^96

500000 = (^) P(1 + 0.^1208596 1 + 0. 085

P = 500000

= 253916.^41

Depreciation

A =^ P1 -^ R)N

Fu = Pv^1 -^ R)w

Example

F = 1040(1 -^0. 18)

= (^) S470. 21

470.^21

1040 X^ (^100) = 45.^2 %

EXPONENTIAL AND LOGARITHMS

Exponential functions

Exponetial functions are^ function where the^ variable^ is^ the^ exponent

Examples

f(x) = 2x f(x) = 5x -^2 f(x) = g2x +^1

f (x) means^ function of

f(x) = (^) y

summary

An asympote is a line that a curve appoaches but never by getting closer and closer but never touches

The (^) graph (^) y = -a^ is^ the reflection (^) of the (^) graph y = a on the (^) x-axis

The graph y = - is the reflection of the graph y = a " on the y-accis

The graph y = -q-^ is^ the^ reflection of the graph y =^ a"^ on^ they-assis and^ -acc is

y =^ a

• y = a" j Y
• (^) &

y =^ -^ q^ Yy^ =^ -^ a

• x

L!

The transformation yat has from its parent function shifts it to the left horizontally

The transformation ya has from its parent function shifts it to the right horizontally

y =^ 2x^ +^1

2 y^ =^ zy = 2x^ -^1

-^ *^ /

Y &

The transformation (^) y =^ at^ has (^) from its (^) parent function shifts it to the (^) up (^) vertically The transformation (^) y =^ a"-I^ has^ from its (^) parent function shifts it^ to the down (^) vertically 3

y =^2 -^1

I E^ &

new the value increases so does the y value from^ having an^ exponential^ equation

new the value decreases so does the y value we^ know^ y =^ 5xy^ =^ -^5 y^ =^ z

• y =^ -^5 -

zin +=

when a^70 the^ y in +=^ (1^ ,^ 5)^ 1in) for simple (^) graphs ( - (^) 115) (^) ( - 11 - 5) ( - (^1) , 5) (-1,^ - 5) when a <0 the^ Yin +=^ -^ y^ =^ a y =^ a^ =^ C

logarithms

y =^ roga)( =^ a^?

The basea 'must^ be positive and^ not^ equal to

10928 =^3 -0^ =^23

To use a^ calculator^ we^ are^ restricted^ to^2 types

log ,^ 04d^ only^ logio^ can^ be^ used^ on^ a^ calculator

I (^1094)

10 ge logna + ural

In 4

short nand Examples

52 = 25 ↓ log5 25 = 2

1095125/54 =^125

54 = 53

= 3

log laws

law (^1) logg)) +^ loggy =^ loga(((y

Example

109 , 04 +^1091025

= 10g , 0100 = log

= 2

law2 loga-logay = loga (

Example

1093243 - logz

= 10gz9 = 10gy

= 2

law3 loga(" =^ nloga

Example

210963 +^210962 =^ 10gg3^ +^ log,

10969 +^ log

109636 =^ 10g)

= 2

Examples

(1094 +^ logg, 10) - Slogc100 - logg20)

=^109640

-1095 = 10938 = 310932 = 10962

given (^) 10g (^) , of =^0. 8451

109 , 0 700

= 10910(7x100)

= 10g107 + 10g , 0100

= 0.^8451 +^2 = 2. 8451 4m +^1 I

= 82 taking logio^ of^ both^ sides

= (m + 1) log , 04 = 00g10)s'a

m + 1 = - 10g ,^ 0)82

m +^1 =^ -^1.^75 m (^) = - 2. 75

log graphs

y =^ 10)^ "I^ is^ n^ inverse (^) function y =^ 10g , (^0)!

EQUATIONS AND FURTHER TRIG

Trigonmetry

It used^ to^ find the missing angle/side in^ right angle and^ non

• right angled triangled triangled^ triangled^ triangles · hypothen use (^) Pythagoras therom ·

E

a2 (^) + b = c2

RULES-SOH CA TO A

adjacent BEARINGS -compass (^) bearings

• (^) True (^) bearings (3 (^) fig (^) bearings ( True (^) bearings
• (^) It is (^) measuring by north in (^) a clockwise direction It (^) always has^3 figs

e

= ↑^600 060 ° T

j

X

compass (^) bearings

It always starts from the north^ or^ south^ assis^ and^ measured

turning toward^ the^ east^ or^ west^ N

N3150 N No

o

N45 °^ E^

1300 x 2 W2708^ Egg0 H (^) N I genjon sigg^ S135 % E 3 The (^) bearing from y from 360°^ -^50 =^3108 E 3 is^1300

. What is the x^ from^ y =^3100

bearings from^ y

co-interior (^) = 500 180 -^130 = (^50)

SIN AND^ COS^ RULES

When (^) trying to^ find

=

an

angle

when trying to find

a

length

• C^

the side is^ always opposite

of it's^ angle x

A C

B

to solve (^) for

Cost angles

a2 (^) = 32 + c -^ 2bc COSA to^ solve^ for leng +^ hS Supplementary (^) angles

• Il^ is^ postive^ in^ Q1 Q2(s) (^) Q1(A) Only sin^ is^ postive^ in^ Q2

1800 jo Only tan^ is postive in^ Q3

Only cos^ is (^) postive in (^) Q4 Qu( +)^ Q4(c) 2700

Exact values

2 special (^) triangles (^30 450) Sin 30 = (^) + tan (^30) : 2 5 ↑^

E

I

= ' (^) I

(

,

60 =^5

EQUATIONS

Examples

2x +^12 =

xx

4(2x + (^) 12) =^ 7(2x +^ 5) 8)x + 48 =^ 2(x^ +^35

13 =^ 13x

x =^1

=

2x -^5

3)+^4

3x +^ 4(2x +^ 3)^ =^ yx^ -^ 1(ax-^ 5)

6x2 + 9x + 8x +^12 =^ Gx)^ - 15x -^ 2x + 5

17x + 12 =^ - 17x +^5

34() =^ -^7 x (^) = Inqualities · (^) = ,

0 =^3 , 4

Examples = (^1) 7x - (^70214) *

*

(^7) 717x[84 (^11 12)

11 [12

• (^2) x + (^70) D
• a(x -^7 - 9 -^7 H - 27 - 2 The (^) sign reverses when -^3 6
• multiplying (^) by a (^) negative

dividing t x412 - taking the^ reciprocal^ of

both sides