Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
Further Algebraic Operations
Skill
Express algebraic fractions with a single denominator, for example:
2
x
3
x
+
4
x + 3
−
5
x + 2
1
x
a
x
3
x + 1
2
x − 1
3
b
y
3
+
x
1
x − 2
−
−
a
x
−
4
x + 3
2
1
+
2
Achieved ?
3
a + 4
4
x + 2
+
4
x
x + 1
Simplify algebraic expressions by cancelling, for example:
−
a
3b
× 2
b
a
5p 2
8
÷
p
2
ab 6
a 3b 2
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Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
3t
s2
×
t
2s
Simplify algebraic expressions by factorising, for example:
3y 2 − 6y
y 2 + y − 6
2x + 2
(x + 1)2
(2x + 5)2
(2x − 1)(2x + 5)
3x − 5
(x − 5)2
Know that to transpose (aka change the subject) a formula for a
specified variable means rearranging an equation in which that
variable is the only quantity on the LHS of the equation
Change the subject of a formula, for example:
st
q
for s
P = R 2b − 5
for R
r = 3p + 2t
for p
r =
y = ax
m =
2
for x
+ 5
3x + 2y
for x
p
2
p = q + 2r
for r
x
+ a = b
c
K =
m 2n
p
p = q +
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for x
for m
a
for a
2
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Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
A = ½h (a + b)
for h
2
A = 4π r
for r
P = 2(L + B )
for L
Know that any number or variable raised to the power of 0 is,
by convention, equal to 1, for example:
30
a
= 1
0
= 1
Know that any number or variable raised to the power of 1 equals
that same number or variable. For example:
91 = 9
r
1
= r
Know that the reciprocal of a number or variable equals 1 divided by
the number or variable, and is written to the power of − 1 :
1
4
4 −1 =
1
s −1 =
Know that a
1q
s
means taking the q
th
root of a, i.e.:
q
a1 q =
a
p q
Know that a
means raising a to the p th power and then taking the
q th root of the result, or equivalently, taking the q th root of a
and then raising the result to the p th power :
ap q
=
1q
(a )
p
q
=
ap
or
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Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
ap
q
( )
a1 q
=
p
( )
q
=
a
p
Simplify numbers with fractional indices, for example:
82 3
=
13
(8 )
2
3
=
82
3
=
64
= 4
or
82 3
=
2
(8 )
13
2
( )
3
=
8
= 22 = 4
Know the rules of indices :
a p × a q = a p +q
a p ÷ a q = a p −q
q
(a )
p
= a
p
×q = a p q
Know that a 1 2 is often written as a
Simplify expressions using the rules of indices, for example:
a
1
2
× a
5
2
a2
−1
a 2  2a 2 + a 

6x
a
2
3
3

2
÷ 2x
1
2
−2 
 23
− a 3
a


3a 5 × 2a
a2
( )
k8 × k2
−3
m 5 × m −8
m5
m3
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Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
(
)
p 3 p 2 − p −3
Know that a surd is a root of a natural number that
cannot be written as a rational number
Know the rules of surds :
ab
=
a
b
=
a
b
a
b
Use the rules of surds to simplify surds, for example:
24
=
3×8
24
2
=
3 8
=
3 4 ×2
24
2
=
12 =
=
=
3 4 2
4×3
= 2 3
= 2 2 3
Simplify other surds, for example:
40
2
6
2 3 ×
Add and subtract surds such as:
18 −
72
2 +
45 − 2 5
12 + 5 3 −
27
2 5 +
20
−
45
63 +
28
−
7
Know that rationalising a denominator in a fraction means
writing the denominator without a surd, for example:
3
24
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=
1
8
=
8
8
5
=
2
4
St. Machar Academy
Int 2 Checklist (Unit 3)
2
5 − 1
2
=
(
(
Int 2 Checklist (Unit 3)
5 + 1
5 − 1
)(
)
2
5 + 1
)
=
(
5 + 1
4
)
=
5 + 1
2
Rationalise a denominator, for example:
2
3
7
2
12
2
Quadratic Functions
Skill
Achieved ?
Know that a quadratic expression is one of the form:
ax 2 + bx + c
Know that a quadratic equation is an equation involving a variable x
that is squared, and usually an x term and a constant number
Know that a quadratic equation in standard form is written as:
ax
2
+ bx + c = 0
Bring a quadratic equation not in standard form
to one that is in standard form
Know that solving a quadratic equation means finding
values of the variable that satisfy the equation
Know that a quadratic equation may have 0, 1 or 2 solutions
Know that there are 3 techniques for solving a quadratic equation,
Factorisation
Quadratic Formula
Graph
Solve a quadratic equation by factorisation, for example:
7 + 6x − x
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2
= 0
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Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
8x − x 2 = 0
Solve a quadratic equation in standard form by
using the Quadratic Formula :
−b ±
x =
b 2 − 4ac
2a
Solve quadratic equations using the Quadratic Formula, for example:
3x
2
− 2x − 6 = 0
3x
2
+ 3x − 7 = 0
2x
2
+ 3x − 1 = 0
2x
2
+ 4x − 9 = 0
2x
2
+ 7x − 3 = 0
4x
2
− 7x + 1 = 0
2x
2
− 6x − 5 = 0
5x
2
+ 4x − 2 = 0
x
2
+ 5x + 3 = 0
Know that every parabola has a maximum or minimum turning point :
(a > 0, minimum)
(a < 0, maximum)
Find the y – intercept of a parabola
Find the x – intercept(s) of a parabola by solving the
associated quadratic equation in standard form
Recognise quadratics of the form y = kx 2 and y = k (x + a)
where a and b are integers, from their graphs
Given the graph of quadratic of the form y = kx
M Patel (August 2011)
7
2
2
+ b,
and one point on
St. Machar Academy
Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
the graph, find the value of k
Know that a parabola has a line of symmetry (parallel to the y – axis)
through the turning point with equation x = constant
State the equation of the symmetry axis of a parabola whose
equation is of the form y = k (x + a) 2 + b when k = ± 1
and a and b are integers
Given the maximum or minimum turning point on the graph of a
parabola of the form y = k (x + a) 2 + b with
k = ± 1 , state the values of a and b
Given a quadratic function and an x – coordinate,
calculate the corresponding y - coordinate
State the coordinates of the maximum or minimum turning point
of a quadratic function of the form y = k (x + a) 2 + b
when k = ± 1 and a and b are integers, for example:
y = 20 − (x − 3) 2
y = (x − 1) 2 − 16
y = (x + 2) 2 − 16
y = 36 − (x − 2) 2
y = (x − 3) 2 − 4
y = 8x − x
2
Given the roots of a quadratic function, find the
coordinates of the turning point
Given the equation of the symmetry axis of a parabola and the
coordinates of a point A not on the symmetry
axis, find the coordinates of another point
B which has the same y - coordinate as A
Further Trigonometry
Skill
Know that the sine, cosine and tangent functions are periodic
Know that the graphs of y = sin x ° and y = cos x ° each have a
period of 360°, amplitude 1, maximum value 1
and minimum value − 1
Know that:
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Achieved ?
St. Machar Academy
Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
sin x
cos x
Know that the graph of y = tan x has a period of 180°,
and no maximum or minimum values
Know that in the graphs of y = a sin bx ° and y = a cos bx °, b
describes how many whole ‘sine or cosine shapes’ fit into a
360
0° to 360° range of x – values; period =
tan x =
b
Given a y – value, use the symmetry of trigonometric graphs
to work out other values
Given the graph of y = a sin bx ° or y = a cos bx ° within a range of
x – values (not necessarily 0 to 360), find the values of a and b
Given the graph of y = tan bx ° within a range of x – values
(not necessarily 0 to 360), find the value of b
Sketch trigonometric graphs of the form y = a sin bx ° or
y = a cos bx ° where a and b are positive integers
for x values satisfying 0 ≤ x ≤ 360
Know that in the graphs of y = sin (x − a) ° or y = cos (x − a) °, a
is a phase shift; a > 0 means the graph of sine or cosine is
shifted a units to the right, a < 0 to the left
Given the graph of y = sin (x − a) ° or y = cos (x − a) °,
state the value of a
Know that a trigonometric equation is an equation
involving a trigonometric function
Solve simple trigonometric equations graphically, for example:
cos x ° = − 0 · 5
(0 ≤ x ≤ 360)
Rearrange a trigonometric equation into one of the 3 forms:
sin x ° = a
( − 1 ≤ a ≤ 1)
cos x ° = a
( − 1 ≤ a ≤ 1)
tan x ° = a
(a is any number)
Solve the above trigonometric equations
for a specified range of x – values
Solve trigonometric equations such as:
4 sin x ° − 1 = 0
(0 ≤ x < 360)
4 tan x ° + 5 = 0
(0 ≤ x ≤ 360)
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Int 2 Checklist (Unit 3)
Int 2 Checklist (Unit 3)
10 + 5 sin t ° = 12 · 5
(0 ≤ x ≤ 180)
2 tan x ° + 7 = 0
(0 ≤ x < 360)
7 sin x ° − 3 = 0
(0 ≤ x ≤ 360)
7 cos x ° − 5 = 0
(0 ≤ x < 360)
8 + 4 sin t ° = 10 · 5
(0 ≤ x ≤ 180)
5 tan x ° − 6 = 2
(0 ≤ x < 360)
4 cos x ° + 3 = 0
(0 ≤ x ≤ 360)
7 sin x ° + 1 = − 5
(0 ≤ x ≤ 360)
15 tan x ° + 1 · 7 = 18 · 4
(0 ≤ x ≤ 90)
2 tan x ° − 3 = 5
(0 ≤ x ≤ 360)
Know the Pythagorean Identity :
sin2 x ° + cos2 x ° = 1
Simplify trigonometric expressions such as:
tan x ° cos x °
cos3 x
1 − sin2 x
Know the meaning of trigonometric identity
Prove trigonometric identities such as:
1 − cos2 A
cos2 A
= tan2 A
tan x ° cos x ° = sin x °
sin3 x ° + sin x ° cos2 x ° = sin x °
sin2 A
1 − sin2 A
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= tan2 A
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