MATH PI NOTES 2024
Asia Yarlett
,
, QI IN PAPER
QUADRATIC FORMULA : K = - bl (b) -
Lac (or EQN on
Calc)
2a
·
"Show that the roots are real and rational" >
- solve for A delta
, Not v
FACTORISING : SUM 3 DIFF OF CUBES :
E
( #21
( RUL
=
S
=
take to
INEQUALITIES : one side k6h
-
-
inequality sign when E
r
-
swap
-
CVS
Cry :
k =1 g
-
Di aso
I aco
remember
: -
8(k < S
-
1 or <
x2 0x + 1620 REAL NUMBER SYSTEM
ey
~
-
.
+
(R + n)(k + 4) > 0
-
4
* IN Natural 1 : 2 ; 3 ;...; a
: RtRinf -
4 No Whole Oil ; 2 ;...; a
a ...; -1
*
Integers : 0 ; 1: c
-
...
SURD EQ : -
square both sides 9 Rational 0 333 0 25 , ,
- check both answers ! # Irrational it 2 35408
, ...
·
2 c =
non-real "F2
(2)5132 12232 :
2 123 when even: Ans
multiplying
-
w/ reciporacle and numerator
=
is
EXPONENT LAWS
EXPONENTS : ·
very big exponent (1002 + 2100 x0 = 1
. common factor
(2)998
=
17 -m
c m
surd bottom
atno
on
tre re
·
: rationalise +
+ r ... umxn = um + n
Conjugate
(um)n = cmn
HIGHER ORDER TIPS :
(m :
C
Any Interger multiplied by even number will be even
-
-
Log laws (see pg with inverse graphs) = "Tem
-
know number system to understand P
My
"
"My =
Y .
mus
Min Mm + m
, FUNCTIONS ,
INVERSES LOGARITHMS
,
FUNCTIONS RECAP
STRAIGHT LINE
+ y
-
m
+ c
y ma
=
gradient
mco m> 0 m is undefined m = 0
Doman Ce ER
To ↳ D
:
1 2
D : HER
EIR R:
y 2
Range Y
=
:
PARABOLA > know nature of rool
VERTICAL SHIFT
TURNING PT FORM :
: shifted up
STANDARD FORM :
U-INT FORM :
HORIZONTAL SHIFT
q>0
shild on a
= Value q horizontal
Stretch ?
re-intercepts
↑ y-int ↑
a(a-p q
↑ &
bx a(x
y
=
y
=
ax + + c y u, )(x (z)
-
= -
SHAPE S >
- A 0 S:
.
. c = a
↳ AOS :
halfway between C-ints
a-value R-value
" from 0 graph TP
=
&O - as a Incr : narrower
↳
.
& O as a dear .
From O :
graph narrower
TPu
(a = 0 - result in straight line)
TPx
f(x) =
YO
CER
Domain :
ifalo Y
NOTE :
:y It O
Range no -ints
no solut
Do
HYPERBOLA HYPERBOLA IN S F .
:
look like denominator
+ (u) get constant by changing to
= -
··
#
aso :
SHAPE /QUADRANTS)
r
S
d
+
y
=
x -
p
& VERTICAL SHIFT
↓ line Is HORIZONTAL ASUMPTOTE
as
y q
=
HORIZONTAL SHIFT
as line U =
p IS VERTICAL ASUMPTOTE
↳ AOS u= p
:Y=KPRO
: sub in cords of asps are
.
.
(P : q)
y = R .: Swap by
x=
Y y+ C
REFLECTION OVER LINE =
Ce +C :
Domain : REIR R + P
, sub into eq
-
in
original co-ords .
to cords of relection
RER y get
Range :
, q
, n
EXPONENTIAL don't confuse
y be
= vs
y y =
SHAPE not asymptote
asymptote HORIZONTAL
aso above SHIFT
If exponent is a variable
aso below asymptote ↑
↑ VERTICAL SHIFT
p
bu
-
&
a +
y &
= as line q IS HORRONTAL ASUMPTOTE
y
=
.
↓
SHAPE
b> 1 (whole #) :
away asymptore
b> 0 < b < /
0 < b < 1 (fraction) ... towards asymptote
away towards
smaller than 1
(b >0 ; b 70
(1 exponential)]
a 70
Fine Endeer.
as will result in str line = 1: no
above
.
= (
= # with
Note : b = any a neg . exponent will always
result I n a
positive #/ fraction ao
-
U RULE OF THUMB : When -value is neg , change b into fraction
below
ener incr
-
Domain : HEIR NOTE : from h(r) =
3-4 to g()
=
27 3 .
x
33 3
-
.g
e
ac
. =
* remember
.
:ya
Range x+ 3
-
no exclude = 3
*
asymprose in standard
form
=
3-24
Range questions : g(u) is him) shilked
units
-
3 to the right
TRANSLATIONS 3 REFLECTIONS
I
-
↑ >
- horizontal Shift (translation)
x -
(+ P)
f(c-p) shifts f(x) p units to the right
f(re + P) shift flue) punits to the Fear
x -
( P)
-
translations
Y
f(u) +
q
-> Vertical Shift (translation)
V
M
f(x)
f(x) + q shifts fas a units up &
V
M
f(x) =
f(u) -
& shifts fas a units Downd
(reflections ( Y
I
f( x)
-
f(x)
f(
yaV
=
x) is a reflection of graph flsel about the -as
↳ a-values change sign
f(r) reflection of flsl about the - Gain (ay)
is a
graph
=
a
reflections
(k ; -
y)
↳ y-values change sign
n
, GRAPH INTERPRETATION
* see Nature of rools
length vertical line :
AB =
Yeop- Ybotrom
length horizontal line :
CD =
Bright-break
f(u)
I
ra a
eg .
:
Max. length AB between C3D :
anywhere
AB =
Stopyou
g(u) =
-
2u + 6 -
(2x2 -
8x + 6)
zre2 >
-
derive iP formula to
+ 6 use get a
=
=
or
e sub In ce-values to get y-value
TIPS FOR FUNCTIONS :
length =
negative E
·
a function is positive above R-axis ; negative O below r-axis
f(u) =
g(r)
:
Where flies above or on 9
Where is one graph positive =- one is
negative ; g(
f :
or fiel =
0 = 0 and g(x) 0 - = under and e values where
graph equals Zero
·
f(ul g(u).
>
0 : where graphs both positive EX0 =
# or both
negative E x 0 =
0
·
f(u) g(k) .
<
0 :
where one graph is positive and one is
negative *
C :
E
-guppankofIntersens
f(u) .
·
> 0 :
y .
r > 0
ly-values &x-values
graph - where It has
asymptotes
·
·
Average gradient : m = ByYo-YoSegur
f'(m) g(r) .
> O : look at gradient of fal and y-vales of glns
·
f" (n) g(u) :
look & concavity Kubis)
, FUNCTIONS
1. ONE-TO-ONE FUNCTION
Di
1 : 1
r-value :
y-value
>
-
single y-value for particular r-value
.
2 MANY-TO-ONE FUNCTION
·.
1 : I
U-value :
y-value
i
>
- more than one -value for a particular y-value
I function can't have more than one
y-value for
every ce-value
HORIZONTAL
Step
# I VERTICAL LINE TEST LINE TEST
To test if function . To represented
a
graph determine type of function
·
is a
·
·
vertical line 11 y-axis (only
·
If
graph passes vertical line test)
if vertical line touches than If horizontal line touches graph
graph
·
more
·
:
once :
I function but a relation . -
more than once
many-to-one
,
~
once :
one-to-one
KGrelation
NOTES
-
f(x) f (x)
·
if original is a one-to-one function therefore inverse is a function
,
if
original is a
many-to-one function, therefore Inverse is a relation unless restricted
·
,INVERSE GRAPHS :
·
reflection about the line
y
= ce =
E
Swap and co-ord With eachother Inverse f(r)
every ce
y
-
.
domain and
= f -
(x)
swap range
-
swap horizontal and vertical asymptotes
Ful
-
=
function
y-direct. f(x) =
2x2
f(u)
#n
v
-
A
> -
-direct &
R
-
=
R
y =
u
relation ↓ l
even
Er =
non-real
XD : Xu
D : UER D: REIR D: KEIR D : UEIR
R: y: 0 R R :
YERR R :
YER
R : yo
- -
Inverse of a
many-to-one function (fup)
inverse of a one-to-one function
is a
one-to-many relation (4) is a function
↓
RESTRICTING THE DOMAIN AND RANGE
In order for the inverse of many-to-one functions to become functions ,
the domain
must be restricted from the turning point (of the original)
Y f f
+
y
+
k
-y
=
· & c
D : 220 or D : [0
R :
Y = 0 R :
Y 10
·
I
D : x20 Dice 20
+U
R :
y 30 mi
y=x y=x
.
&
T
&
D: 2 ? / or
D: x =
2
R :
y= R:
y = 2
-
-
it i
D: u= -
2 D : C = -
2
R ? I =
:
Y R :
y
,THE LOGARITHMIC FUNCTION
=( : bu
form to log form bottom
converting exponential E
:
base
argument base exponent Argument FO :
to
positive number raised
b
as any
ax =
; b > O
and b +1 an exponent o negative
- (will always be greater than zero
logbCe
~
D bF1 R30
y =
-
; >O and and eg. 2 = it =
↓ ↓
exponent bottom argument
(base)
·
Inverse of exponential function (one-to-one) is a log function (one-to-one)
a70 >
- above asymptole 9 o >
- above asymptole
O < b < 1 - towards asymptote b> 1 -
away asymptote
an
& -
y =
original ax
y
: =
: Inverse : x = al 2
make "y" subject, Y =
logal
using log ~
y =
logal y
=
logaz
* remember to
incr,
j
exp decr. exclude asymphote as s incr exp incr
as
y
. .
,
In domain and
range questions -
HER
40
Di CER D: 270 D:
X swap asymptotes X
R:
Y > 0 R: YER R:
doman ((u) range G*()
=
LOG LAWS not in curriculum -
higher order
base 10
*
log always k
logly
=
eg <
(
.
·
logb (ky)
=
loggle +
logby
logsa
common mistakes :
logb (g) =
logple-logbY logu xlogy logcety)
log , an
=
nlogpa log(- y
logbe- Glogiu logy
:
lok =
y
y a
by loga ,
=
* When
working out inequalities and -
logazo (negative E)
ga
make sure to swap sign If
bottom
be
loge=0 argumentn
· is
,
-x =
logaY R2 + 0 >
- undelined (asymptote)
, NUMBER PATTERNS , SEQUENCES AND SERIES
h ↓
set of ordered numbers terms of a
(progressions)
adding sequence
① ARITHMETIC (LINEAR) constant difference (d)
SEQUENCE @ ; a +; a + 2 ;....; a + (n 2(d -
; a + (n -
1)d
a+ (1 1)d
- a + (2- 1)d a + 23 -
1)d a + ((n 1)d
-
1) -
=
a =
a + d a + 2d
a + n 2)d-
① >
- C up in Is
e
general term
↳ position of nth term UEN
in = a + (n -
1)(d)
↳ first term L constant first diff .
d = in -
Tn-1
d Tz -
T 1: Ts - T2
ARITHMETIC MEAN
=
e position :
T3 .: n =
3 if aib ; C , ...
is an
Th Th ⑭ arithmetic sequence ,
then
ai(a + d) j (a +
2d) a + 2
9
b =
9
-X-13
e
.
9
.
; 2
> value
- of 3rd term : T3 =
13
+ 4 + 4
>
- constant diff (d) .
SERIES : Sn = T1 + Tz + Tz +... + Tn -
1 + Tn
data sheet & last term known
Sn = [2a + (n-1)d] .
Sn =
E(a +
1]
9 + (a + d) +
(a + 2d) + ...
Tu
or a + (n -
1)d
-
PROOF # & (second last) Clast)
#m I =
#
① Sn A t (a + d) + (a (n 2)d)
=
+... + -
+ (a + (n -
1)d)
T #m = 1 P2 #
I
② Sn = (a + (n -
1(d) + (a + (n -
2(d) + ...
+ (a + d) + A
a + d + a + (n 2)d
-
① + ② 2a + d + dn - 2d
2a + dn - d
(sn =
(2a + (n -
1)a) + (2mm)d) + .2a.n.
1(d) + -
+ (2a + (n -
1)d)
2Sn = n(2a + (n -
1)d] >
- n number of terms
Sn = (2a + (n -
1)d]
Asia Yarlett
,
, QI IN PAPER
QUADRATIC FORMULA : K = - bl (b) -
Lac (or EQN on
Calc)
2a
·
"Show that the roots are real and rational" >
- solve for A delta
, Not v
FACTORISING : SUM 3 DIFF OF CUBES :
E
( #21
( RUL
=
S
=
take to
INEQUALITIES : one side k6h
-
-
inequality sign when E
r
-
swap
-
CVS
Cry :
k =1 g
-
Di aso
I aco
remember
: -
8(k < S
-
1 or <
x2 0x + 1620 REAL NUMBER SYSTEM
ey
~
-
.
+
(R + n)(k + 4) > 0
-
4
* IN Natural 1 : 2 ; 3 ;...; a
: RtRinf -
4 No Whole Oil ; 2 ;...; a
a ...; -1
*
Integers : 0 ; 1: c
-
...
SURD EQ : -
square both sides 9 Rational 0 333 0 25 , ,
- check both answers ! # Irrational it 2 35408
, ...
·
2 c =
non-real "F2
(2)5132 12232 :
2 123 when even: Ans
multiplying
-
w/ reciporacle and numerator
=
is
EXPONENT LAWS
EXPONENTS : ·
very big exponent (1002 + 2100 x0 = 1
. common factor
(2)998
=
17 -m
c m
surd bottom
atno
on
tre re
·
: rationalise +
+ r ... umxn = um + n
Conjugate
(um)n = cmn
HIGHER ORDER TIPS :
(m :
C
Any Interger multiplied by even number will be even
-
-
Log laws (see pg with inverse graphs) = "Tem
-
know number system to understand P
My
"
"My =
Y .
mus
Min Mm + m
, FUNCTIONS ,
INVERSES LOGARITHMS
,
FUNCTIONS RECAP
STRAIGHT LINE
+ y
-
m
+ c
y ma
=
gradient
mco m> 0 m is undefined m = 0
Doman Ce ER
To ↳ D
:
1 2
D : HER
EIR R:
y 2
Range Y
=
:
PARABOLA > know nature of rool
VERTICAL SHIFT
TURNING PT FORM :
: shifted up
STANDARD FORM :
U-INT FORM :
HORIZONTAL SHIFT
q>0
shild on a
= Value q horizontal
Stretch ?
re-intercepts
↑ y-int ↑
a(a-p q
↑ &
bx a(x
y
=
y
=
ax + + c y u, )(x (z)
-
= -
SHAPE S >
- A 0 S:
.
. c = a
↳ AOS :
halfway between C-ints
a-value R-value
" from 0 graph TP
=
&O - as a Incr : narrower
↳
.
& O as a dear .
From O :
graph narrower
TPu
(a = 0 - result in straight line)
TPx
f(x) =
YO
CER
Domain :
ifalo Y
NOTE :
:y It O
Range no -ints
no solut
Do
HYPERBOLA HYPERBOLA IN S F .
:
look like denominator
+ (u) get constant by changing to
= -
··
#
aso :
SHAPE /QUADRANTS)
r
S
d
+
y
=
x -
p
& VERTICAL SHIFT
↓ line Is HORIZONTAL ASUMPTOTE
as
y q
=
HORIZONTAL SHIFT
as line U =
p IS VERTICAL ASUMPTOTE
↳ AOS u= p
:Y=KPRO
: sub in cords of asps are
.
.
(P : q)
y = R .: Swap by
x=
Y y+ C
REFLECTION OVER LINE =
Ce +C :
Domain : REIR R + P
, sub into eq
-
in
original co-ords .
to cords of relection
RER y get
Range :
, q
, n
EXPONENTIAL don't confuse
y be
= vs
y y =
SHAPE not asymptote
asymptote HORIZONTAL
aso above SHIFT
If exponent is a variable
aso below asymptote ↑
↑ VERTICAL SHIFT
p
bu
-
&
a +
y &
= as line q IS HORRONTAL ASUMPTOTE
y
=
.
↓
SHAPE
b> 1 (whole #) :
away asymptore
b> 0 < b < /
0 < b < 1 (fraction) ... towards asymptote
away towards
smaller than 1
(b >0 ; b 70
(1 exponential)]
a 70
Fine Endeer.
as will result in str line = 1: no
above
.
= (
= # with
Note : b = any a neg . exponent will always
result I n a
positive #/ fraction ao
-
U RULE OF THUMB : When -value is neg , change b into fraction
below
ener incr
-
Domain : HEIR NOTE : from h(r) =
3-4 to g()
=
27 3 .
x
33 3
-
.g
e
ac
. =
* remember
.
:ya
Range x+ 3
-
no exclude = 3
*
asymprose in standard
form
=
3-24
Range questions : g(u) is him) shilked
units
-
3 to the right
TRANSLATIONS 3 REFLECTIONS
I
-
↑ >
- horizontal Shift (translation)
x -
(+ P)
f(c-p) shifts f(x) p units to the right
f(re + P) shift flue) punits to the Fear
x -
( P)
-
translations
Y
f(u) +
q
-> Vertical Shift (translation)
V
M
f(x)
f(x) + q shifts fas a units up &
V
M
f(x) =
f(u) -
& shifts fas a units Downd
(reflections ( Y
I
f( x)
-
f(x)
f(
yaV
=
x) is a reflection of graph flsel about the -as
↳ a-values change sign
f(r) reflection of flsl about the - Gain (ay)
is a
graph
=
a
reflections
(k ; -
y)
↳ y-values change sign
n
, GRAPH INTERPRETATION
* see Nature of rools
length vertical line :
AB =
Yeop- Ybotrom
length horizontal line :
CD =
Bright-break
f(u)
I
ra a
eg .
:
Max. length AB between C3D :
anywhere
AB =
Stopyou
g(u) =
-
2u + 6 -
(2x2 -
8x + 6)
zre2 >
-
derive iP formula to
+ 6 use get a
=
=
or
e sub In ce-values to get y-value
TIPS FOR FUNCTIONS :
length =
negative E
·
a function is positive above R-axis ; negative O below r-axis
f(u) =
g(r)
:
Where flies above or on 9
Where is one graph positive =- one is
negative ; g(
f :
or fiel =
0 = 0 and g(x) 0 - = under and e values where
graph equals Zero
·
f(ul g(u).
>
0 : where graphs both positive EX0 =
# or both
negative E x 0 =
0
·
f(u) g(k) .
<
0 :
where one graph is positive and one is
negative *
C :
E
-guppankofIntersens
f(u) .
·
> 0 :
y .
r > 0
ly-values &x-values
graph - where It has
asymptotes
·
·
Average gradient : m = ByYo-YoSegur
f'(m) g(r) .
> O : look at gradient of fal and y-vales of glns
·
f" (n) g(u) :
look & concavity Kubis)
, FUNCTIONS
1. ONE-TO-ONE FUNCTION
Di
1 : 1
r-value :
y-value
>
-
single y-value for particular r-value
.
2 MANY-TO-ONE FUNCTION
·.
1 : I
U-value :
y-value
i
>
- more than one -value for a particular y-value
I function can't have more than one
y-value for
every ce-value
HORIZONTAL
Step
# I VERTICAL LINE TEST LINE TEST
To test if function . To represented
a
graph determine type of function
·
is a
·
·
vertical line 11 y-axis (only
·
If
graph passes vertical line test)
if vertical line touches than If horizontal line touches graph
graph
·
more
·
:
once :
I function but a relation . -
more than once
many-to-one
,
~
once :
one-to-one
KGrelation
NOTES
-
f(x) f (x)
·
if original is a one-to-one function therefore inverse is a function
,
if
original is a
many-to-one function, therefore Inverse is a relation unless restricted
·
,INVERSE GRAPHS :
·
reflection about the line
y
= ce =
E
Swap and co-ord With eachother Inverse f(r)
every ce
y
-
.
domain and
= f -
(x)
swap range
-
swap horizontal and vertical asymptotes
Ful
-
=
function
y-direct. f(x) =
2x2
f(u)
#n
v
-
A
> -
-direct &
R
-
=
R
y =
u
relation ↓ l
even
Er =
non-real
XD : Xu
D : UER D: REIR D: KEIR D : UEIR
R: y: 0 R R :
YERR R :
YER
R : yo
- -
Inverse of a
many-to-one function (fup)
inverse of a one-to-one function
is a
one-to-many relation (4) is a function
↓
RESTRICTING THE DOMAIN AND RANGE
In order for the inverse of many-to-one functions to become functions ,
the domain
must be restricted from the turning point (of the original)
Y f f
+
y
+
k
-y
=
· & c
D : 220 or D : [0
R :
Y = 0 R :
Y 10
·
I
D : x20 Dice 20
+U
R :
y 30 mi
y=x y=x
.
&
T
&
D: 2 ? / or
D: x =
2
R :
y= R:
y = 2
-
-
it i
D: u= -
2 D : C = -
2
R ? I =
:
Y R :
y
,THE LOGARITHMIC FUNCTION
=( : bu
form to log form bottom
converting exponential E
:
base
argument base exponent Argument FO :
to
positive number raised
b
as any
ax =
; b > O
and b +1 an exponent o negative
- (will always be greater than zero
logbCe
~
D bF1 R30
y =
-
; >O and and eg. 2 = it =
↓ ↓
exponent bottom argument
(base)
·
Inverse of exponential function (one-to-one) is a log function (one-to-one)
a70 >
- above asymptole 9 o >
- above asymptole
O < b < 1 - towards asymptote b> 1 -
away asymptote
an
& -
y =
original ax
y
: =
: Inverse : x = al 2
make "y" subject, Y =
logal
using log ~
y =
logal y
=
logaz
* remember to
incr,
j
exp decr. exclude asymphote as s incr exp incr
as
y
. .
,
In domain and
range questions -
HER
40
Di CER D: 270 D:
X swap asymptotes X
R:
Y > 0 R: YER R:
doman ((u) range G*()
=
LOG LAWS not in curriculum -
higher order
base 10
*
log always k
logly
=
eg <
(
.
·
logb (ky)
=
loggle +
logby
logsa
common mistakes :
logb (g) =
logple-logbY logu xlogy logcety)
log , an
=
nlogpa log(- y
logbe- Glogiu logy
:
lok =
y
y a
by loga ,
=
* When
working out inequalities and -
logazo (negative E)
ga
make sure to swap sign If
bottom
be
loge=0 argumentn
· is
,
-x =
logaY R2 + 0 >
- undelined (asymptote)
, NUMBER PATTERNS , SEQUENCES AND SERIES
h ↓
set of ordered numbers terms of a
(progressions)
adding sequence
① ARITHMETIC (LINEAR) constant difference (d)
SEQUENCE @ ; a +; a + 2 ;....; a + (n 2(d -
; a + (n -
1)d
a+ (1 1)d
- a + (2- 1)d a + 23 -
1)d a + ((n 1)d
-
1) -
=
a =
a + d a + 2d
a + n 2)d-
① >
- C up in Is
e
general term
↳ position of nth term UEN
in = a + (n -
1)(d)
↳ first term L constant first diff .
d = in -
Tn-1
d Tz -
T 1: Ts - T2
ARITHMETIC MEAN
=
e position :
T3 .: n =
3 if aib ; C , ...
is an
Th Th ⑭ arithmetic sequence ,
then
ai(a + d) j (a +
2d) a + 2
9
b =
9
-X-13
e
.
9
.
; 2
> value
- of 3rd term : T3 =
13
+ 4 + 4
>
- constant diff (d) .
SERIES : Sn = T1 + Tz + Tz +... + Tn -
1 + Tn
data sheet & last term known
Sn = [2a + (n-1)d] .
Sn =
E(a +
1]
9 + (a + d) +
(a + 2d) + ...
Tu
or a + (n -
1)d
-
PROOF # & (second last) Clast)
#m I =
#
① Sn A t (a + d) + (a (n 2)d)
=
+... + -
+ (a + (n -
1)d)
T #m = 1 P2 #
I
② Sn = (a + (n -
1(d) + (a + (n -
2(d) + ...
+ (a + d) + A
a + d + a + (n 2)d
-
① + ② 2a + d + dn - 2d
2a + dn - d
(sn =
(2a + (n -
1)a) + (2mm)d) + .2a.n.
1(d) + -
+ (2a + (n -
1)d)
2Sn = n(2a + (n -
1)d] >
- n number of terms
Sn = (2a + (n -
1)d]
