Proof
Proof By Contradiction Proving a statement by showing the opposite can't be true
:
.
Proof By Deduction Logical argument through logical statements and deductions.
:
Proof By Counterexample :
Use an example to show a statement isn't always truee.
Proof By Exhaustion :
Proving a statement
using smaller , simpler scenarios & proving .
each
Algebra and functions
Discriminants Quadratics
:
Solving :
:
a) + 612 + C
62-49) 0 -
No real Roots CTS : ②
Quadratic formula :
OR
62 49) 0 I repeated root
a(x (a) 162- nac
=
6 +
- -
+ + c 0 ) = m(x x ,)
-
(y y
= -
-
,
x =
2a
62 -uac > O - 2 Distinct roots
↑
used to find the
turning point
* substitution * Elimination
To solve Simultaneous Equations , make lory) the subject of an equation and substitute into another a make the <
Cory) coefficient have the same coefficient from the other , and subtract(only for linear) .
↳Solutions show where the INTERSECT
graphs .
factor Theorem f(x)) is f(p) D2-p) is factor off
It polynomial then 0 &
: =
a ,
a
Sketching Polynomials
&
C intercepts by 0
"I
- =
in even , and o , anco
M
a + a x a ,x + x y intercept y do/constant term
↓ ↓
+
y
y
-
= a anx
:
+... +
, ,
Turning Point to Differentiation
If n= odd , and ot" , anso
h
= (x -
P,)(x -
Pc) ... (x -
Pr)
Reciprocal functions proportionality ~ /INVERSEL How to solve modulus
↓ Square both sides then solve
·
↓ neveodd
.
yxx
,
yax
f(x) =
a ·
Graph & find intersecting points &
y = kx
y
=
x identify a values corresponding to
7gative
(a , n - 0)
= y,x
=
y232
+ ve values of the functions .
Modulus Graphs Defining functions Inverse functions
Domain for "DC) their fix)'s
w
:
Inputs (x) values values
y
f(x)s
=
,
Range :
outputs (y) forf"(2) ,
their values =
f(x)'s y values
·
nu reflected
One-to-one ↳ Reflected IN y =
C
If()l < in the
: each
axis
xdy is unique
Partial fractions
f((x1) reflected Many-to-one : In a
- in they domain () of f , there's f(xC) A B C
axis = + +
also another possible domain (ax + b) ((x + d)2 ax + b a +d (x + d)2
(x2) f(x) =
f(x2) * When should
Improper algebraic division
·
,
be used
First .
Graph Transformations
y
=
f(x) + 6 -
up
by b //
y
=
f(x) -> scale
vertically by 6 //y =
- fD) >
-
reflect in axis
y f
(s) + 6)
= -
LEfTbya//y =
flas) - Scale
horizontally by "a /y =
f(x) >
- reflect in yaxis
↳ order Horizontal
:<
scaling & reflections 3 vertical scaling & reflections
2 Horizontal translations 4 vertical translations
,Coordinate Geometry
Equation ofA Circle Parametric Equations
p(t)
Y single
x describes xay in terms of t.
(y q)
=
(x b)
-
+ -
= r2
& convert to this
form via CTS .
y
=
9 (t) equation is called a cartesion equation
.
Sequences and Series
(b) -
The number of ways Binomial Expansion
of choosing items
(a + bx)" =
an + (2)a" bx + (2)an - b-x +... + (n)an rbx
-
+... + baxh
from a set of .
n
↳ can be used to answer questions such as -
Expand 12-34
n !
-
r ! (n-r) !
General Binomial Expansion
ONLY VALID WHEN KCKI -(EG : 12 3)
-
""
e 2(1 +
titl +
( yaxs -y)) z)
-
+ ...
) converging sequence :
-
-
him Un for
a ll
n(a)x (g)
44 - )
(a + bx)" a1 r + 1)
. (n -
x
ji
= + +... +
-
+...
Arithmetic Sequence Arithmetic Series
Un =
Un-i + d Sn =
=(2a + (n -
1)d]
=
a + (n -
1)d
Geometric
Sequence Geometric Series
common Ratio
Convergent Geometric Series
Un &
Un r 1)
marn If Irk1
arn-
=
run-
-
Sa
,
= =
Un-1 Sn = =
it's
1
r -
1 -
r
convergent
Trigonometry 90r
N
sindo
Converting To Radians :
Converting DegreesTo : Arc Length Sector Area
S
A
1x, y
00
Y
188
= r -
E
T
Orad =
Odegx , 80 Odeg =
Orad x 180 S= - A =
x
-tands o
Area of A
Triangle Sine Rule Cosine Rule
A ↳ C
A = absinc I
sin B
=
sinc
a2 = b2 + c -
26c osA
sin A
Small Angle Approximation irig symmetry periodicity Relating cos & sin
Asymptotes
Singe Sinx Sin(x = 2 x) Sinc cos( *
(C) tanc
nen
Sin (x-x) Sinc
-
= = = 2
cost = 1 -
[02 COS 122 -
x) =
COSIC (os(x2 = 2a) = COSI (osx = Sin (
*
2 -x)
(2k + 1)π
=
x
tano Sin(-x) = -
sinc tan(x = x) =
tanc 2
cos( >() COSSC
-
- =
tan 1-x) = -
tank
same for sec , coses
and eat
, ·
Reciprocals
cose((2
I 1 v
m
=
Secs =
Sins COSx
·
undefined at x = kx
undefined at
12k + 11
-M
x=
2
&
I
(Otx
·
=
tanx
undefined at x = k2
tanx = cot l -
<
Inverse functions
trig Identities
arcsin sin"x & rcCOSC Cos " x arctanx tan"x
in
= = =
tanf =
langle - langle 0= 0= a) langle -0
Sino COS G = 1
m
+
~ ↳
-
lotf =
10sO
SinG
=
coseCeT
reciprocal
Seco
Trig
Identities
V coseco = 1 + cot
seco = 1 + tan@
-
-
I
Simplifying Trig Expressions : Example :
Addition Angle formula
simplify to Rsin(OIC) simplify 3sinO + 4 cost to the form
Sin(A = B) = sinAcosB = COSAsinB ↳
R =
+ b2
Rsin(0 + (C)
Rsin(0 + 1) RsinOcos
Ros
= +
COS/A = B) = COSACOSB SinAsinB ↳ tan" (i) -
c =
↳ 34 =
" 4
to make 3 sing to make 4 cos8
anAItan B
tan(AIB) =
1 tan Atan B simplify to RCOS(OIC
-
>
R =
b2 )
.
3 + 4 = RCOSC) + Rsince =
R(cos + since)
3 + 42 = R2(cos" < + Sin = c) =
R (1)
=
↳ c =
tan" (f)
~
R2 =
32 + 42 = 25
R =
c =
5---- -
tanceSin
a = tan" (b) =
53 1
.
3 sinO 4 cos E
formula
Angle
+
Double Trigequations 1)
=
Ssin 10 53
solve + .
-
Sin (2A) = 2SinACOSA EG : find solutions to sin (2x +
*
2) =
2 1 -
>)
2 = x -
cos(2A) = cos A-sin
? ?
A .
1 Adjust the domain - 2x
+
2a + 7/2
2COSPA-1
=
?
- +
= 1 -
2 sin A
. +an A
2
solvable state sin" (2)
tan (2A) =
.
2
Rearrange to a 2x + = =
=
1 - tan ? A
I
3 and
.
Use
symmetry (x + =
-
periodicity to find
solutions.
x=
52
-
Proof By Contradiction Proving a statement by showing the opposite can't be true
:
.
Proof By Deduction Logical argument through logical statements and deductions.
:
Proof By Counterexample :
Use an example to show a statement isn't always truee.
Proof By Exhaustion :
Proving a statement
using smaller , simpler scenarios & proving .
each
Algebra and functions
Discriminants Quadratics
:
Solving :
:
a) + 612 + C
62-49) 0 -
No real Roots CTS : ②
Quadratic formula :
OR
62 49) 0 I repeated root
a(x (a) 162- nac
=
6 +
- -
+ + c 0 ) = m(x x ,)
-
(y y
= -
-
,
x =
2a
62 -uac > O - 2 Distinct roots
↑
used to find the
turning point
* substitution * Elimination
To solve Simultaneous Equations , make lory) the subject of an equation and substitute into another a make the <
Cory) coefficient have the same coefficient from the other , and subtract(only for linear) .
↳Solutions show where the INTERSECT
graphs .
factor Theorem f(x)) is f(p) D2-p) is factor off
It polynomial then 0 &
: =
a ,
a
Sketching Polynomials
&
C intercepts by 0
"I
- =
in even , and o , anco
M
a + a x a ,x + x y intercept y do/constant term
↓ ↓
+
y
y
-
= a anx
:
+... +
, ,
Turning Point to Differentiation
If n= odd , and ot" , anso
h
= (x -
P,)(x -
Pc) ... (x -
Pr)
Reciprocal functions proportionality ~ /INVERSEL How to solve modulus
↓ Square both sides then solve
·
↓ neveodd
.
yxx
,
yax
f(x) =
a ·
Graph & find intersecting points &
y = kx
y
=
x identify a values corresponding to
7gative
(a , n - 0)
= y,x
=
y232
+ ve values of the functions .
Modulus Graphs Defining functions Inverse functions
Domain for "DC) their fix)'s
w
:
Inputs (x) values values
y
f(x)s
=
,
Range :
outputs (y) forf"(2) ,
their values =
f(x)'s y values
·
nu reflected
One-to-one ↳ Reflected IN y =
C
If()l < in the
: each
axis
xdy is unique
Partial fractions
f((x1) reflected Many-to-one : In a
- in they domain () of f , there's f(xC) A B C
axis = + +
also another possible domain (ax + b) ((x + d)2 ax + b a +d (x + d)2
(x2) f(x) =
f(x2) * When should
Improper algebraic division
·
,
be used
First .
Graph Transformations
y
=
f(x) + 6 -
up
by b //
y
=
f(x) -> scale
vertically by 6 //y =
- fD) >
-
reflect in axis
y f
(s) + 6)
= -
LEfTbya//y =
flas) - Scale
horizontally by "a /y =
f(x) >
- reflect in yaxis
↳ order Horizontal
:<
scaling & reflections 3 vertical scaling & reflections
2 Horizontal translations 4 vertical translations
,Coordinate Geometry
Equation ofA Circle Parametric Equations
p(t)
Y single
x describes xay in terms of t.
(y q)
=
(x b)
-
+ -
= r2
& convert to this
form via CTS .
y
=
9 (t) equation is called a cartesion equation
.
Sequences and Series
(b) -
The number of ways Binomial Expansion
of choosing items
(a + bx)" =
an + (2)a" bx + (2)an - b-x +... + (n)an rbx
-
+... + baxh
from a set of .
n
↳ can be used to answer questions such as -
Expand 12-34
n !
-
r ! (n-r) !
General Binomial Expansion
ONLY VALID WHEN KCKI -(EG : 12 3)
-
""
e 2(1 +
titl +
( yaxs -y)) z)
-
+ ...
) converging sequence :
-
-
him Un for
a ll
n(a)x (g)
44 - )
(a + bx)" a1 r + 1)
. (n -
x
ji
= + +... +
-
+...
Arithmetic Sequence Arithmetic Series
Un =
Un-i + d Sn =
=(2a + (n -
1)d]
=
a + (n -
1)d
Geometric
Sequence Geometric Series
common Ratio
Convergent Geometric Series
Un &
Un r 1)
marn If Irk1
arn-
=
run-
-
Sa
,
= =
Un-1 Sn = =
it's
1
r -
1 -
r
convergent
Trigonometry 90r
N
sindo
Converting To Radians :
Converting DegreesTo : Arc Length Sector Area
S
A
1x, y
00
Y
188
= r -
E
T
Orad =
Odegx , 80 Odeg =
Orad x 180 S= - A =
x
-tands o
Area of A
Triangle Sine Rule Cosine Rule
A ↳ C
A = absinc I
sin B
=
sinc
a2 = b2 + c -
26c osA
sin A
Small Angle Approximation irig symmetry periodicity Relating cos & sin
Asymptotes
Singe Sinx Sin(x = 2 x) Sinc cos( *
(C) tanc
nen
Sin (x-x) Sinc
-
= = = 2
cost = 1 -
[02 COS 122 -
x) =
COSIC (os(x2 = 2a) = COSI (osx = Sin (
*
2 -x)
(2k + 1)π
=
x
tano Sin(-x) = -
sinc tan(x = x) =
tanc 2
cos( >() COSSC
-
- =
tan 1-x) = -
tank
same for sec , coses
and eat
, ·
Reciprocals
cose((2
I 1 v
m
=
Secs =
Sins COSx
·
undefined at x = kx
undefined at
12k + 11
-M
x=
2
&
I
(Otx
·
=
tanx
undefined at x = k2
tanx = cot l -
<
Inverse functions
trig Identities
arcsin sin"x & rcCOSC Cos " x arctanx tan"x
in
= = =
tanf =
langle - langle 0= 0= a) langle -0
Sino COS G = 1
m
+
~ ↳
-
lotf =
10sO
SinG
=
coseCeT
reciprocal
Seco
Trig
Identities
V coseco = 1 + cot
seco = 1 + tan@
-
-
I
Simplifying Trig Expressions : Example :
Addition Angle formula
simplify to Rsin(OIC) simplify 3sinO + 4 cost to the form
Sin(A = B) = sinAcosB = COSAsinB ↳
R =
+ b2
Rsin(0 + (C)
Rsin(0 + 1) RsinOcos
Ros
= +
COS/A = B) = COSACOSB SinAsinB ↳ tan" (i) -
c =
↳ 34 =
" 4
to make 3 sing to make 4 cos8
anAItan B
tan(AIB) =
1 tan Atan B simplify to RCOS(OIC
-
>
R =
b2 )
.
3 + 4 = RCOSC) + Rsince =
R(cos + since)
3 + 42 = R2(cos" < + Sin = c) =
R (1)
=
↳ c =
tan" (f)
~
R2 =
32 + 42 = 25
R =
c =
5---- -
tanceSin
a = tan" (b) =
53 1
.
3 sinO 4 cos E
formula
Angle
+
Double Trigequations 1)
=
Ssin 10 53
solve + .
-
Sin (2A) = 2SinACOSA EG : find solutions to sin (2x +
*
2) =
2 1 -
>)
2 = x -
cos(2A) = cos A-sin
? ?
A .
1 Adjust the domain - 2x
+
2a + 7/2
2COSPA-1
=
?
- +
= 1 -
2 sin A
. +an A
2
solvable state sin" (2)
tan (2A) =
.
2
Rearrange to a 2x + = =
=
1 - tan ? A
I
3 and
.
Use
symmetry (x + =
-
periodicity to find
solutions.
x=
52
-
