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Edexcel AS and A level Mathematics
Pure Mathematics
Year 1 /AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson, Bronwen/uni00A0Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith, Geoο¬ /uni00A0Staley, Robert Ward-Penny, Dave Wilkins11 β 19 PRO... | [
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Edexcel AS and A level Mathematics
Pure Mathematics
Year 1 /AS
Series Editor: Harry Smith
Authors: Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson, Bronwen/uni00A0Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Pledger, Harry Smith, Geoο¬ /uni00A0Staley, Robert Ward-Penny, Dave Wilkins11 β 19 PRO... | [
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iiContents
Overarching themes iv
Extra online c
ontent vi
1 Algebraic e
xpressions 1
1.1 Index law
s 2
1.2 Expanding brack
ets 4
1.3 Factorising 6
1.4 Negative and fractional indic
es 9
1.5 Surds 12
1.6 Rationalising denominators 13
Mixed ex
ercise 1 15
2 Quadratics 18
2.1 Solving quadratic equations ... | [
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iiiContents
8.5 Binomial estimation 167
Mixed ex
ercise 8 169
9 Trigonometric r
atios 173
9.1 The cosine rul
e 174
9.2 The sine rule 179
9.3 Areas o
f triangles 185
9.4 Solving triangle pr
oblems 187
9.5 Graphs of sine, c
osine and tangent 192
9.6 Trans
forming trigonometric graphs 194
Mixed ex
ercise ... | [
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0... |
ivOverarching themes
The following three overarching themes have been fully integrated throughout the Pearson Edexcel
AS and A level Mathematics series, so they can be applied alongside your learning and practice.
1. Mathematical argument, language and proof
β’ Rigorous and consistent approach throughoutβ’ Notation box... | [
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0.... |
vOverarching themes
Every few chapters a Review exercise
helps you consolidate your learning with lots of exam-style questionsEach section begins
with explanation and key learning points
Step-by-step worked
examples focus on the key types of questions youβll need to tackleExercise questions are
carefully graded so ... | [
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viExtra online content
Whenever you see an Online box, it means that there is extra online content available to support you.
SolutionBank
SolutionBank provides a full worked solution for
every question in the book.
Download all the solutions
as a PDF or quickly fi nd the solution you need online Extra online content... | [
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viiExtra online content
Access all the extra online content for FREE at:
www.pearsonschools.co.uk/p1maths
You can also access the extra online content by scanning this QR Code:
GeoGebra interactives
Explore topics in more detail,
visualise problems and consolidate your understanding with GeoGebra-powered interactiv... | [
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viiiPublished by Pearson Education Limited, 80 Strand, London WC2R 0RL.
www.pearsonschoolsandfecolleges.co.uk Copies of official specifications for all Pearson qualifications may be found on the website:
qualifications.pearson.com
Text Β© Pearson Education Limited 2017
Edited by Tech-Set Ltd, GatesheadTypeset by Tech-... | [
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1
Algebraic expressions
After completing this chapter you should be able to:
β Multiply and divide integer po
wers β pages 2β3
β Expand a single term over brackets and collect like
terms
β pages 3β4
β Expand the product of two or three expressions β pages 4β6
β Factorise linear, quadratic and simple cubic expre... | [
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-0.06750431656837463,
0.019... |
2
Chapter 1
1.1 Index laws
β You can use the laws of indices to simplify powers of the same base.
β’ am Γ an = am + n
β’ am Γ· an = am β n
β’ (am)n = amn
β’ (ab)n = anbn
Example 1
Example 2
Expand these expressions and simplify if possible:
a β3x
(7x β 4) b y2(3 β 2y3)
c 4x
(3x β 2x2 + 5x3) d 2x (5x + 3) β 5(2x + 3)Simplif... | [
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-0.04340866208076477,
0.0018016294343397021,
0.022351857274770737,
0.017754772678017616,
-0.022316208109259605,
-0.03964490815997124,
0.06053663045167923,
-0.... |
3Algebraic expressions
a β3x(7xΒ β 4 ) =Β β21x2Β +Β 12 x
b y2(3Β βΒ 2y3) =Β 3 y2Β βΒ 2y5
c 4x(3xΒ βΒ 2 x2Β +Β 5 x3)
=Β 12 x2Β βΒ 8 x3Β +Β 20 x4
d 2x(5xΒ +Β 3 )Β βΒ 5(2 xΒ +Β 3)
=Β 10 x2Β +Β 6 xΒ βΒ 10 xΒ βΒ 15
=Β 10 x2Β βΒ 4 xΒ βΒ 15
a x7 + x4 _______ x3 = x7 ___ x3 + x4 ___ x3
=
x7 β 3Β + x4 β 3 = x4Β + x
b 3x2 β 6x5 __________ 2... | [
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-0.0252936240285635,
-0.038605... |
4
Chapter 1
1.2 Expanding brackets
To find the product of two expressions you multiply each term in one expression by each term in the
other expression.
(x + 5)(4x β 2y + 3)x Γ
5 Γ= x(4x β 2y + 3) + 5(4x β 2y + 3)= 4x
2 β 2xy + 3x + 20x β 10y + 15
= 4x2 β 2xy + 23x β 10y + 15Multiplying each of the 2 terms in the firs... | [
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0.01709219068288803,
0.00... |
5Algebraic expressions
c (x β y)2
= (x β y)(x β y)
=
x2 β xy β xy + y2
= x2 β 2xy + y2
d (x + y)(3x β 2 y β 4)
= x(3x
β 2y β 4) + y (3x β 2 y β 4)
= 3x2 β 2xy β 4 x + 3 xy β 2 y2 β 4y
= 3x2 + xy β 4 x β 2 y2 β 4y
a x(2x + 3)(x β 7)
= (2x2 + 3 x)(x β 7)
= 2
x3 β 14 x2 + 3 x2 β 21x
= 2
x3 β 11 x2 β 21x
b ... | [
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0.0203... |
6
Chapter 1
1.3 Factorising
You can write expressions as a product of their factors.
β Factorising is the opposite of expanding
brack
ets.4x(2x + y)
(x + 5)3
(x + 2y)(x β 5y)= 8x2 + 4xy
= x3 + 15x2 + 75x + 125
= x2 β 3xy β 10y2Expanding brackets
FactorisingExpand and simplify ( x + y )4. You can use the binomial expa... | [
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0.0... |
7Algebraic expressions
An ex pression in the form x2 β y2 is
called the difference of two squares.Notation= (x + 3)(2x β 1)β A quadratic expression has the form
ax2 + bx + c where a, b and c are real
numbers and a β 0.
To factorise a quadratic expression:
β’Find two fact
ors of ac that add up to b
β’Rewrite the
b... | [
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0... |
8
Chapter 1
Example 8
Factorise completely:
a x3 β 2x2 b x3 β 25x c x3 + 3x2 β 10xb x2 + 6 x + 8
= x2 + 2 x + 4 x + 8
= x(x
+ 2) + 4( x + 2)
= (x
+ 2)( x + 4)
c 6x2 β 11 x β 10
= 6x2 β 15 x + 4 x β 10
= 3x(2x
β 5) + 2(2 x β 5)
= (2 x
β 5)(3 x + 2)
d x2 β 25
= x2 β 52
= (x + 5)( x β 5)
e 4x2 β 9 y2
= 22x2 ... | [
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-0.069... |
9Algebraic expressions
Write 4x4 β 13x2 + 9 as the product of four linear factors.Challenge2 Factorise:
a x2 + 4x b 2x2 + 6x c x2 + 11x + 24
d x2 + 8x + 12 e x2 + 3xΒ β 40 f x2 β 8x + 12
g x2 + 5x + 6 h x2 β 2xΒ β 24 i x2 β 3xΒ β 10
j x2 +Β xΒ β 20 k 2x2 + 5xΒ + 2 l 3x2 + 10x β 8
m 5x2 β 16xΒ + 3 n 6x2 β 8x β 8
o 2x2 + 7xΒ β 1... | [
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10
Chapter 1
Example 9
Simplify:
a x 3 ___ x β3 b x1
2 Γ x32
c (x3)23
d 2x1.5Β Γ·Β 4xβ0.25 e 3 ββ―______ 125 x 6 f 2 x 2 β x _______ x 5
a x 3 ____ x β3 = x3 β (β3) = x6
b x1
2 Γ x3
2 = x1
2 ξ±Β 32 = x2
c (x3)23 =Β x3 ξ³Β 23 =Β x2
d 2x1.5Β ξ΄Β 4 xβ0.25 = 1 __ 2 x1.5Β β (β0 ... | [
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-0.0740... |
11Algebraic expressions
1 Simplify:
a x3 Γ· xβ2 b x5 Γ· x7 c x 3 _ 2 Γ x 5 _ 2
d (x2 ) 3 _ 2 e (x3 ) 5 _ 3 f 3x0.5 Γ 4xβ0.5
g 9 x 2 _ 3 Γ· 3 x 1 _ 6 h 5 x 7 _ 5 Γ· x 2 _ 5 i 3x4 Γ 2xβ5
j ββ―__
x Γ 3 ββ―__
x k ( ββ―__
x )3 Γ ( 3 ββ―__
x... | [
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-0.008... |
12
Chapter 1
1.5 Surds
If n is an integer that is not a square number, then any multiple of ββ―__
n is called a surd.
Examples of surds are ββ―__
2 , ββ―___ 19 and 5 ββ―__
2 .
Surds are examples of irrational numbers.
The decimal expansion of a surd is never-ending and never repeats, for example
... | [
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-0.0... |
13Algebraic expressions
Expand and simplify if possible:
a ββ―__
2 (5 β ββ―__
3 ) b (2 β ββ―__
3 )(5 + ββ―__
3 ) Example 13
a ββ―__
2 (5 β ββ―__
3 )
= 5 ββ―__
2 β ββ―__
2 ββ―__
3
= 5 ββ―__
2 β ββ―__
6
b (2 β
ββ―__
3 )(5 + ββ―__
3 )
= 2(5
+ ββ―__
3 ) β ββ―__
... | [
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14
Chapter 1
Rationalise the denominator of:
a 1 ___ ββ―__
3 b 1 ______ 3 + ββ―__
2 c ββ―__
5 + ββ―__
2 _______ ββ―__
5 β ββ―__
2 d 1 ________ (1 β ββ―__
3 )2 Example 14
a 1 ___ ββ―__
3 = 1 Γ ββ―__
3 ________ ββ―__
3 Γ ββ―__
3
= ββ―__
3 ___ 3
b... | [
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-0.0334... |
15Algebraic expressions
1 Simplify:
a 1 ___ ββ―__
5 b 1 ____ ββ―___ 11 c 1 ___ ββ―__
2 d ββ―__
3 ____ ββ―___ 15
e ββ―__ 12 ____ ββ―__ 48 f ββ―__
5 ____ ββ―___ 80 g ββ―___ 12 _____ ββ―____ 156 h ββ―__
7 ____ ββ―___ 63
2 Rationa
lise the denom... | [
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0.01935826614499092,
-0.00... |
16
Chapter 1
5 Factorise these expr
essions completely:
a 3x2 + 4x b 4y2 + 10y c x2 + xy + xy2 d 8xy2 + 10x2y
6 Factorise:
a x2 + 3x + 2 b 3x2 + 6x c x2 β 2x β 35 d 2x2 β x β 3
e 5x2 β 13x β 6 f 6 β 5 x β x2
7 Factorise:
a 2x3 + 6x b x3 β 36x c 2x3 + 7x2 β 15x
8 Simplify:
a 9x3 Γ· 3xβ3 b ( 4 3 _ 2 ) ... | [
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17Algebraic expressions
20 Solve the equation 8 + x ββ―___ 12 = 8x ___ ββ―__
3
Give y
our answer in the form a ββ―__
b where a and b are integers. (4 marks)
21 A rectangle has a length of (1 + ββ―__
3 ) cm and area of ββ―___ 12 cm2.
Calculate the width of the rectangle in cm.
Express your an... | [
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0.0016906982054933906,
-0.03... |
18
Quadratics
After completing this chapter you should be able to:
β Solve quadratic equations using fact
orisation, the quadratic
formula and completing the square β pages 19 β 24
β Read and use f(x) notation when working with
functions β pages 25 β 27
β Sketch the graph and find the turning point of a quadratic ... | [
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0.025523... |
19Quadratics
2.1 Solving quadratic equations
A quadratic equation can be written in the form ax2 + bx + c = 0, where a, b and c are real constants,
and a β 0. Quadratic equations can have one, two, or no real solutions.
β To solve a quadratic equation by factorising:
β’ Writ
e the equation in the form ax2 + bx + c = 0... | [
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0.03265965357422829,
-0.0... |
20
Chapter 2
In some cases it may be more straightforward to solve a quadratic equation without factorising.
Example 2
Solve the following equations
a (2x
β 3)2 = 25 b (x β 3)2 = 7
a (2x β 3)2 = 25
2x
β 3 = Β±5
2x = 3 Β±
5
The
n either 2x = 3 + 5 β
x = 4
or 2x = 3 β
5 β x = β 1
The solutions are x = 4 and x ... | [
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0.0473746582865715,
-0.0657033696770668,
0.033453140407800674,
0.02269... |
21Quadratics
x = β (β7) Β± β ______________ (β7) 2 β 4 (3) (β1) _______________________ 2 Γ 3
x =
7 Β± β _______ 49 + 12 _______________ 6
x =
Β 7 Β± β ___ 61 ________ 6 Β
The
n x =
7 + β ___ 61 ________ 6 or x = 7 β β ___ 61 _______ 6
Or x
= 2.47 (3 s.f.) or ... | [
-0.02378137782216072,
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0.012003909796476364,
0.01792... |
22
Chapter 2
Given that x is positive, solve the equation
1 __ x + 1 _____ x + 2 = 28 ____ 195 Challenge Write the equation in the form
ax2 + bx + c = 0 before using the quadratic
formula or factorising.Hint
2.2 Completing the square
It is frequently useful to rewrite quadratic expressions by complet... | [
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0.02437373623251915,
-0.053487... |
23Quadratics
3x2 + 6 x + 1
= 3(x2 + 2x) + 1
= 3(( x + 1)2 β 12) + 1
= 3(x + 1)2 β 3 + 1
= 3(x + 1)2 β 2
So p = 3, q = 1 and r = β 2.Example 5
Write 3x2 + 6x + 1 in the form p(x + q)2 + r, where p, q and r are integers to be found.
1 Complete the square for the e
xpressions:
a x2 + 4x b x2 β 6x c x2 β 16x d x2 + x e ... | [
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0.07446865737438202,
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0.10164717584848404,
-0.05958917737007141,
0.0121842036023736,
0.0020413044840097427,
-0.005... |
24
Chapter 2
Solve the equation 2x2 β 8x + 7 = 0. Give your answers in surd form.Example 7
2x2 β 8 x + 7 = 0
x2 β 4 x + 7 __ 2 = 0
x2 β 4 x = β 7 __ 2
(x β
2)2 β 22 = β 7 __ 2
(x β
2)2 = β 7 __ 2 + 4
(x
β 2)2 = 1 __ 2
x β
2 = Β± ββ―__
1 __ 2
x =
2 Β± 1 ___ ββ―__
2
So th
... | [
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0.02562318556010723,
0.060167163610458374,
-0.07456247508525848,
-0.017832111567258835,
-0.071272... |
25Quadratics
2.3 Functions
A function is a mathematical relationship that maps each value of a set of inputs to a single output.
The notation f(x) is used to represent a function of x.
β The set of possible inputs for a function is called the domain.
3DomainR ange
7
β7
2f(β7) = 49f(7) = 49f(3) = 9
f( 2) = 29
49
49... | [
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-0.04918083921074867,
0.01122002862393856,
-0.006405067630112171,
0.014039368368685246,
0.101796... |
26
Chapter 2
c (x + 3)2 > 0
So the minimum value of f( x) is β14.
This occurs when ( x + 3)2 = 0,
so when x = β 3A squared value must be greater than or equal to 0.
Find the roots of the function f(x) = x6 + 7x3 β 8, xΒ β β .Example 10
f(x) = 0
x6 + 7x3 β 8 = 0
(x3)2 + 7( x3) β 8 = 0
(x3 β 1)( x3 + 8) = 0
So x3 = 1 ... | [
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-0.00505... |
27Quadratics
4 The functions p and q are giv
en by p(x) = x2 β 3x and q(x) = 2x β 6, x β β .
Find the two v
alues of x for which p(x) = q(x).
5 The functions f and g are gi
ven by f(x) = 2x3 + 30x and g(x) = 17x2, Β x β β .
Find the three v
alues of x for which f(x) = g(x).
6 The function f is defined as f(x
) = x... | [
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-0.02854674495756626,
-0... |
28
Chapter 2
As a = 1 is positive, the graph has a
shape and a minimum point.
When x = 0, y = 4, so the graph crosses
the y-axis at (0, 4).
When y = 0,
x2 β 5 x + 4 = 0
(x β 1)( x β 4) = 0
x = 1 or x = 4, so the graph crosses the x-axis at (1, 0) and (4, 0).
Completing the square:
x
2 β 5 x + 4 = (x β 5 _ 2 ) ... | [
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0.00949497427791357,
0.040153957... |
29Quadratics
As a = β 2 is negative, the graph has a
shape and a maximum point.
When x = 0, y = β 3, so the graph
crosses the y -axis at (0, β 3).
When y = 0,β2x
2 + 4x β 3 = 0
Using the quadratic formula,
x = β4 Β± β _____________ 4 2 β 4 (β2) (β3) ____________________ 2 Γ (β2)
x =
β4 Β± β __... | [
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0.03615638613700867,
0.05059301108121872,
-0.07606087625026703,
-0.012585122138261795,
0.0228... |
30
Chapter 2
1 Sketch the gra
phs of the following equations. For each graph, show the coordinates of the point(s)
where the graph crosses the coordinate axes, and write down the coordinate of the turning point
and the equation of the line of symmetry.
a y =
x2 β 6x + 8 b y = x2 + 2x β 15 c y = 25 β x2 d y = x2... | [
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0.02153606154024601,
0.03756258264183998,
-0.08058635145425797,
0.05030027776956558,
0.03649017587304115,
-0.04662029817700386,
-0.022744959220290184,
-0.013... |
31Quadratics
You can use the discriminant to check the shape of sketch graphs.
Below are some graphs of y = f(x) where f(x) = ax2 + bx + c.
a . 0
y
x O
y
x O y
x O
b2 β 4ac . 0 b2 β 4ac = 0 b2 β 4ac , 0
Two distinct real roots One repeated r oot No real roots
a
, 0
y
x O
y
x O
y
x O
Find the range of values... | [
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0.00960448570549488,
0.002563233021646738,
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0.007404150906950235,
-0.03229108825325966,
-0.08342619240283966,
-0.05215401574969292,
0.0336... |
32
Chapter 2
1 a Calcula
te the value of the discriminant for each of these five functions:
i f(x)
= x2 + 8x + 3 ii g(x) = 2x2 β 3x + 4 iii h(x) = βx2 + 7x β 3
iv j(x)
= x2 β 8x + 16 v k(x ) = 2x β 3x2 β 4
b Using your answ
ers to part a, match the same five functions to these sketch graphs.
i
x Oy ii
x Oy iii ... | [
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0.025897622108459473,
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-0.06933324784040451,
-0.059752628207206726,
0.02... |
33Quadratics
A spear is thrown over level ground from the top of a tower.
The height, in metres, of the spear above the ground after t seconds is modelled by the function:
h(t) = 12.25 + 14.7t β 4.9t2, t > 0
a Interpret the meaning of the constant ter
m 12.25 in the model.
b After how many seconds does the spear hit t... | [
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-0.04870618134737015,
-0.... |
34
Chapter 2
1 The diagram sho
ws a section of a suspension bridge carrying a road over water.
The height of the cables above water level in metres can be modelled by the function
h(x)Β =Β 0.000 12x2 + 200, where x is the displacement in metres from the centre of the bridge.
a Interpret the meaning of the constant ter
... | [
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-0.005397175904363394,
0.021432947367429733,
-0... |
35Quadratics
The total revenue, Β£r, can be calculated by multiplying the number of tickets sold by the price of
each ticket. This can be written as r = p(M β 1000p).
b Rearrange r into the f
orm A β B(p β C)2, where A, B and C are constants to be found. (3 marks)
c Using your answ
er to part b or otherwise, work out... | [
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0.025579236447811127,
-0.054870445281267166,
-0.07496178895235062,
-0.07548219710588455,
0.... |
36
Chapter 2
7 Given tha
t for all values of x:
3x2 + 12x + 5 = p(x + q)2 + r
a find the values of
p, q and r. (3 marks)
b Hence solve the equation 3
x2 + 12x + 5 = 0. (2 marks)
8 The function f is defined as f(x
) = 22x β 20(2x) + 64, x β β .
a Write f(x
) in the form (2x β a)(2x β b), where a and b are real const... | [
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0.0011862489627674222,
-0.0637127235531807,
-0.07562480121850967,
-0.0742... |
37Quadratics
1 To solve a quadratic equation by factorising:
β Write the equation in the f
orm ax2 + bx + c = 0
β Factorise the l
eft-hand side
β Set each factor equal to z
ero and solve to find the value(s) of x
2 The solutions of the equation ax2 + bx + c = 0 where a β 0 are given by the formula:
x = βb Β± ββ―____... | [
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0.002535366453230381,
-0.09215235710144043,
0.011073657311499119,
0.0140... |
38
Equations and
inequalities
After completing this chapter you should be able to:
β Solve linear simultaneous equations using elimination or
substit
ution β pages 39 β 40
β Solve simultaneous equations: one linear and one quadratic
β pages 41 β 42
β Interpret algebraic solutions of equations graphically β pages... | [
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0.02276924... |
39
Equations and inequalities
3.1 Linear simultaneous equations
Linear simultaneous equations in two unknowns have one set of values that will make a pair of
equations true at the same time.
The solution to this pair of simultaneous equations is x = 5, y = 2
x + 3y = 11 (1)
4x
β 5y = 10 (2)
β Linear simultaneous eq... | [
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0.0589090... |
40
Chapter 3
Example 2
1 Solve these simultaneous equations by elimination:
a 2x β
y = 6 b 7x +
3y = 16 c 5x
+ 2y = 6
4x
+ 3y = 22 2x +
9y = 29 3x β
10y = 26
d 2x β
y = 12 e 3x β
2y = β6 f 3x
+ 8y = 33
6x
+ 2y = 21 6x +
3y = 2 6x =
3 + 5y
2 Solve these simultaneous equa
tions by substitution:
... | [
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0.05312203988432884,
0.00486091710627079,
0.04899703711271286,
0.04045356065034866,
-0.05981069... |
41
Equations and inequalities
3.2 Quadratic simultaneous equations
You need to be able to solve simultaneous equations where one equation is linear and one is quadratic.
To solve simultaneous equations involving one linear equation and one quadratic equation, you need
to use a substitution method from the linear equa... | [
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0.04656057059764862,
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42
Chapter 3
3 Solve the simultaneous equa
tions, giving your answers in their simplest surd form:
a x β
y = 6 b 2x
+ 3y = 13
xy =
4 x2 + y2 = 78
4 Solve the simultaneous equa
tions:
x +
y = 3
x2 β 3y = 1 (6 marks)
5 a By eliminating
y from the equations
y = 2 β 4x
3x2 + xy + 11 = 0
show that x2 β 2x β 11 = ... | [
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43
Equations and inequalities
a
β2 β/four.ss01 /four.ss01 8 2 6y
x123/four.ss01
β1
β2
β3
β/four.ss012x + 3 y = 8
3x β y = 23O
b The solution is (7, β2) or x = 7, y = β2.The point of intersection is the solution to the
simultaneous equations
2x + 3y = 8
3x β y = 23
This solution matches the algebraic solution to
the ... | [
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0.03243115171790123,
0.... |
44
Chapter 3
β For a pair of simultaneous equations that produce a quadratic equation of the form
ax2 + bx + c = 0:
β’ b2 β 4ac > 0 β’ b2 β 4ac = 0 β’ b2 β 4ac < 0
two real solutions one real solution no real solutions
Example 6
The line with equation y = 2x + 1 meets the curve with equation kx2 + 2y + (k β 2) = 0 ... | [
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0.031596507877111435,
-0.0059... |
45
Equations and inequalities
1 In each case:
i draw the gr
aphs for each pair of equations on the same axes
ii find the coordinates of
the point of intersection.
a y =
3x β 5 b y =
2x β 7 c y =
3x + 2
y =
3 β x y =
8 β 3x 3x
+ y + 1 = 0
2 a Use graph paper to draw accurately the graphs of 2y = 2x + 11 a... | [
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0.06572654843330383,
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0.025881079956889153,
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-0.0253805760294199,
0.012420202605426311,
0.04552... |
46
Chapter 3
3.4 Linear inequalities
You can solve linear inequalities using similar methods to those for solving linear equations.
β The solution of an inequality is the set o
f all real numbers x that make the inequality true.
Example 7
Find the set of values of x for which:
a 5x
+ 9 > x + 20 b 12 β 3
x < 27
c 3(x
... | [
0.016399066895246506,
0.06043170019984245,
0.09353082627058029,
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0.00316721317358315,
-0.0984192043542862,
0.034346237778663635,
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0.06643901765346527,
-0.012915395200252533,
-0.06189008429646492,
0.019199756905436516,
0.009743... |
47
Equations and inequalities
Example 8
Find the set of values of x for which:
a 3x
β 5 < x + 8 and 5x > x β 8
b x β
5 > 1 β x or 15 β 3x > 5 + 2x.
c 4x
+ 7 > 3 and 17 < 11 + 2x.
a 3x β 5 < x + 8 5x > x β 8
2x β 5 < 8 4x >
β 8
2x < 13 x >
β2
x < 6.5
β/four.ss01 β2 0 /four.ss01 2 6 8
x < 6.5
x > β2
So the requ... | [
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-0.07961013913154602,
0.048516351729631424,
0.018879... |
48
Chapter 3
3.5 Quadratic inequalities
β To solve a quadratic inequality:
β’ Rearr
ange so that the right-hand side of the inequality is 0
β’ Solve the corresponding quadratic equation to find the critical values
β’ Sketch the graph of the quadratic function
β’ Use your sketch to find the required set of values.
The sketc... | [
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0.103... |
49
Equations and inequalities
Example 9
Find the set of values of x for which:
3 β 5x β 2x2 < 0.
3 β 5 x β 2x2 = 0
2x2 + 5 x β 3 = 0
(2x β 1)( x + 3) = 0
x = 1 __ 2 or x = β 3
β3 1
2xy
O
So the required set of values is
x < β3 or x > 1 __ 2 .Multiply by β1 (so itβs easier to factorise).
1 _ 2 and β3 ar... | [
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-0.09216229617595673,
-0.03740864247083664,
0.036... |
50
Chapter 3
Example 11
Find the set of values for which 6 __ x > 2 , x β 0b Solving 12 + 4 x > x2 gives β 2 < x < 6.
Solving 5 x β 3 > 2 gives x > 1.
β/four.ss01β202/four.ss0168
β2 < x < 6
x > 1
The two sets of values overlap where
1 < x < 6.
So the solution is 1 < x < 6.
6 __ x > x
6x
> 2x2
6x β 2x2 > 0
... | [
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-0.02294725552201271,
0.030172277241945267,
-0.... |
51
Equations and inequalities
3 Use set notation to describe the set of v alues of x for which:
a x2 β 7x + 10 < 0 and 3x + 5 < 17 b x2 β x β 6 > 0 and 10 β 2x < 5
c 4x2 β 3x β 1 < 0 and 4(x + 2) < 15 β (x + 7) d 2x2 β x β 1 < 0 and 14 < 3x β 2
e x2 β x β 12 > 0 and 3x + 17 > 2 f x2 β 2x β 3 < 0 and x2 β 3x + 2 > 0
... | [
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0.04395867511630058,
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0.06039924919605255,
0.0393194... |
52
Chapter 3
Example 12
L1 has equation y = 12 + 4x.
L2 has equation y = x2.
The diagram shows a sketch of L1 and L2 on the same axes.
a Find the coordinates of
P1 and P2, the points of intersection.
b Hence write down the solution to the inequality 12
+ 4x > x2.y
x OL1: y = 12 + 4x
L2: y = x2P1
P2
a x2 = 12 + 4 x... | [
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0.0499584823846817,
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0.044180333614349365,
0.03993464633822441,
-0.0017477524233981967,
0.0232989601790905,
0.0... |
53
Equations and inequalities
The sketch shows the graphs of
f(x) = x2 β 4x β 12
g(x) = 6 + 5 x β x2
a Find the coordinates of the points of intersection.
b Fin
d the set of values of x for which f( x) < g( x).
Give your answer in set notation.y
x O
y = g(x)y = f (x)Challenge
All the shaded points in this region sat... | [
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0.04792482778429985,
-0.04281258583068848,
-0.026247045025229454,
-0.040408939123153687,
0.0278... |
54
Chapter 3
β If y
> f(x) or y < f(x) then the curve y = f(x) is not included in the region and is represented
by a dotted line.
β If y > f(x) or y < f(x) then the curve y = f(x) is included in the region and is represented by a
solid line.
Example 13
On graph paper, shade the region that satisfies the inequalities... | [
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0.03958097845315933,
0.0057642157189548016,
0.03810950741171837,
0.0738588273525238,
0.08055... |
55
Equations and inequalities
1 On a coordinate grid, shade the r
egion that satisfies the inequalities:
y > x β 2, y < 4x and y < 5 β x.
2 On a coordinate grid, shade the r
egion that satisfies the inequalities:
x > β1, y + x < 4, 2x + y < 5 and y > β2.
3 On a coordinate grid, shade the r
egion that satisfies the i... | [
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0.05208682641386986,
-0.09709397703409195,
-0.010302322916686535,
0.03112560696899891,
0.07... |
56
Chapter 3
1 2kx
β y = 4
4kx + 3y = β2
are two simultaneous equations, where k is a constant.a
Show that
y = β2. (3 marks)
b Find an expression f
or x in terms of the constant k. (1 mark)
2 Solve the simultaneous equa
tions:
x + 2y = 3
x2 β 4y2 = β33 (7 marks)
3 Given the sim
ultaneous equations
x β 2y = 13x... | [
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-0.005999353714287281,
-0.013044092804193497,
0.03844192996621132,
-0.0... |
57
Equations and inequalities
8 A person throws a ba
ll in a sports hall. The height of the ball, h m, h
x
can be modelled in relation to the horizontal distance from the
point it was thrown from by the quadratic equation:
h = β 3 __ 10 x2 + 5 _ 2 x + 3 _ 2
The hall has a sloping ceiling which can... | [
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0.07510519027709961,
0.04953298717737198,
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0.02139967679977417,
-0.008192496374249458,
-0.05544678866863251,
0.11054953932762146,
0.04... |
58
Chapter 3
1 Find the possible values of k for the quadratic equation 2 kx2 + 5kx + 5k β 3 = 0
to have real roots.
2 A strai
ght line has equation y = 2 x β k and a parabola has equation
y = 3 x2 + 2kx + 5 where k is a constant. Find the range of values of k for which
the line and the parabola do not intersect.Ch... | [
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0.005796846933662891,
-0.006365366745740175,
-0.02673928439617157,
0.025214320048689842,
0.... |
59
Graphs and
transformations
After completing this chapter you should be able to:
β Sketch cubic gr
aphs β pages 60 β 64
β Sketch quartic graphs β pages 64 β 66
β Sketch reciprocal graphs of the form y = a __ x and y = a __ x2 β pages 66 β 67
β Use intersection points of graphs to solve equations β p... | [
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0.04791460931301117,
-0.0443246029317379,
-0.05406191200017929,
-0.04597516357898712,
-0.... |
60
Chapter 4
4.1 Cubic graphs
A cubic function has the form f(x) = ax3 + bx2 + cx + d, where a, b, c and d are real numbers and a is
non-zero.
The graph of a cubic function can take several different forms, depending on the exact nature of the
function.
Oy
x Oy
x Oy
x Oy
x
β If p is a root of the function f( x), then... | [
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0.09336422383785248,
0.039575833827257156,
-0.012838655151426792,
-0.08487225323915482,
0.... |
61Graphs and transformations
So the curve crosses the x -axis at
(0, 0), ( β1, 0) and ( β2, 0).
x β β , y β β
x β ββ, y β β β
β1 1 β2O xyYou know that the curve crosses the x-axis at
(0, 0) so you donβt need to calculate the y-intercept separately.
Check what happens to y for large positive and negative values of x... | [
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0.0377005971968174,
-0.05392921715974808,
-0.001179443788714707,
0.... |
62
Chapter 4
Example 3
Sketch the curve with equation y = (x β 1)(x2 + x + 2).
y = ( x β 1)( x2 + x + 2)
0 = ( x β 1)( x2 + x + 2)
So x = 1 only and the curve crosses the
x-axis at (1, 0).
When x = 0, y = ( β1)(2) = β 2
So the curve crosses the y -axis at (0, β 2).
x β β , y β β
x β ββ, y β β β
y
x O
β21The quadratic... | [
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0.047036632895469666,
0.03328702226281166,
-0.00296430173330009,
0.014659454114735126,
0.04... |
63Graphs and transformations
2 Sketch the curves with the f
ollowing equations:
a y =
(x + 1)2(x β 1) b y = (x + 2)(x β 1)2 c y = (2 β x)(x + 1)2
d y = (x β 2)(x + 1)2 e y = x2(x + 2) f y = (x β 1)2x
g y =
(1 β x)2(3 + x) h y = (x β 1)2(3 β x) i y = x2(2 β x)
j y =
x2(x β 2)
3 Factorise the follo
wing eq... | [
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-0.032204121351242065,
-0.01351320743560791,
-0.03305774927139282,
0.008042678236961365,
0.0123... |
64
Chapter 4
Example 4
Sketch the following curves:
a y =
(x + 1)(x + 2)(x β 1)(x β 2) b y =
x(x + 2)2(3 β x) c y = (x β 1)2(x β 3)2
a y = (x + 1)( x + 2)( x β 1)( x β 2)
0 = ( x + 1)( x + 2)( x β 1)( x β 2)
So x = β 1, β2, 1 or 2
The curve cuts the x -axis at ( β2, 0), ( β1, 0),
(1, 0) and (2, 0).
When x = 0, y... | [
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0.025389019399881363,
-0.037942059338092804,
-0.03609432652592659,
-0.0020098411478102207,
0.04330183193087578,
-0.057650335133075714,
0.037844374775886536,
-0.05502891540527344,
0.05546768009662628,
0.049632325768470764,
-0.027806919068098068,
0.004702458158135414,
... |
65Graphs and transformations
b y = x(x + 2)2(3 β x )
0 = x(x + 2)2(3 β x )
So x = 0, β 2 or 3
The curve cuts the x -axis at (0, 0), ( β2, 0)
and (3, 0)
x β β , y β β β
x β ββ, y β β β
Oy
x β2 3
c y = (x β 1)2(x β 3)2
0 = ( x β 1)2(x β 3)2
So x = 1 or 3
The curve touches the x -axis at (1, 0) and
(3, 0).
When x = 0,... | [
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0.004696281161159277,
-... |
66
Chapter 4
3 The graph of
y = x 4 + bx3 + cx2 + dx + e is shown opposite,
where b, c, d and e are real constants.
a Find the coordinates of
point P. (2 marks)
b Find the values of
b, c, d and e. (3 marks)
4 Sketch the gra
ph of y = (x + 5)(x β 4)(x2 + 5x + 14). (3 marks)E/P
OPy
x32 β2β1
E/P
Consider the disc... | [
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0.060460373759269714,
0.03172998130321503,
-0.06593326479196548,
-0.04839285835623741,
0.0421614... |
67Graphs and transformations
Example 5
Sketch on the same diagram:
a y =
4 __ x and y = 12 ___ x b y = β 1 __ x and y = β 3 __ x c y = 4 __ x2 and y = 10 ___ x2
a
O12
xy =
12
xy =/four.ss01
xy =
/four.ss01
xy =xy
b
c
10
x2 y =10
x2 y =
/four.ss01
x2 y =/four.ss01
x2 y =
x Oy1
xy... | [
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-0.032653696835041046,
-0.04588412120938301,
0.03751184418797493,
-0.... |
68
Chapter 4
4.4 Points of intersection
You can sketch curves of functions to show points of intersection and solutions to equations.
β The x-coordinate(s) at the points of intersection of the curves with equations
y = f(x) and y = g(x) are the solution(s) to the equation f( x) = g( x).
a y
x
CBA
1 3Oy = x(x β 3)
y =... | [
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0.12505751848220825,
0.02858470380306244,
0.00849381648004055,
-0.038004230707883835,
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0.05547834932804108,
0.031587522476911545,
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0.015139855444431305,
0.011190874502062798,
-0.060521893203258514,
-0.0472819060087204,
0.02537... |
69Graphs and transformations
a y
x Oy = x2(3x β a)
ab
xy =b
x
1
3y =
b From the sketch there are only two points
of i
ntersection of the curves. This means
there are only two values of x where
x2 (3x β a) = b __ x
or x2 (3x β a) β b __ x = 0
So this equation has two real solutions.You can sketch curves in... | [
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0.004166868980973959,
-0.04292477294802666,
0.019671732559800148,
0.012028956785798073,
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-0.06092631816864014,
-0.005618855822831392,
... |
70
Chapter 4
a y =
x2, y = x(x2 β 1) b y = x(x + 2), y = β 3 __ x c y = x2, y = (x + 1)(x β 1)2
d y = x2(1 β x), y = β 2 __ x e y = x(x β 4), y = 1 __ x f y = x(x β 4), y = β 1 __ x
g y =
x(x β 4), y = (x β 2)3 h y = βx3, y = β 2 __ x i y = βx3, y = x2
j y = β x3, y = βx(x +... | [
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0.04751107469201088,
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0.024006370455026627,
0.05221424996852875,
-0.03407466039061546,
0.009938026778399944,
-0.0347... |
71Graphs and transformations
11 a Sketch the gra
phs of y = x2(x β 1)(x + 1) and y = 1 _ 3 x3 + 1. (5 marks)
b Find the number of r
eal solutions to the equation 3x2(x β 1)(x + 1) = x3 + 3. (1 mark)E/P
4.5 Translating graphs
You can transform the graph of a function by altering the function. Adding or subtractin... | [
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-0.047871969640254974,
-0.050703033804893494,
0.014110229909420013,
-0.005732616409659386,
-0.04154257848858833,
0.011037252843379974,
0.02051602117717266,
0.018284481018781662,
-0.02727763168513775,
-0.005836542695760727,
0.00... |
72
Chapter 4
c y = x2 + 2
y
x2
OThis is a translation by vector ( 0 2 ) .
Remember to mark on the y-axis intersection.
Example 10
f(x) = x3
g(x) = x(x β 2)
Sketch the following graphs, indicating any points where the curves cross the axes:a
y =
f(x + 1)
b y =
g(x + 1)
a The graph of f( x) is
y
x Oy = f( x) = x... | [
-0.030355287715792656,
0.012230916880071163,
-0.0545712448656559,
-0.08790231496095657,
-0.049356989562511444,
0.060417257249355316,
0.03651876747608185,
-0.018507512286305428,
-0.052644215524196625,
0.09456232935190201,
-0.00821249932050705,
-0.04320719838142395,
0.006469930522143841,
0.0... |
73Graphs and transformations
So the graph of y = g( x + 1) is
y
xy = g( x + 1)
= (x + 1)( x β 1)
1 β1β1O
β When you translate a function, any asymptotes are also translated.
Example 11
Given that h(x) = 1 __ x , sketch the curve with equation y = h(x) + 1 and state the equations of any
asymptotes and intersec... | [
-0.0021682120859622955,
0.05173252895474434,
-0.011672512628138065,
-0.04665299504995346,
-0.10555767267942429,
-0.003565988503396511,
0.06433342397212982,
0.019084274768829346,
-0.027772502973675728,
-0.017244718968868256,
0.01330284122377634,
-0.010055229999125004,
-0.005291614681482315,
... |
74
Chapter 4
1 Apply the f
ollowing transformations to the curves with equations y = f(x) where:
i f(x
) = x2 ii f(x) = x3 iii f(x) = 1 __ x
In each case state the coordina
tes of points where the curves cross the axes and in iii state the
equations of the asymptotes.
a f(x + 2) b f(x) + 2 c f(x
β 1)
d... | [
-0.016780594363808632,
0.054968323558568954,
-0.08192957192659378,
-0.008509675040841103,
0.018274690955877304,
0.05182822793722153,
0.004271028097718954,
0.041193392127752304,
-0.12794603407382965,
-0.015121717937290668,
0.02162160724401474,
-0.07069893181324005,
0.03224297985434532,
0.03... |
75Graphs and transformations
10 a Sketch the gra
ph of y = x2(x β 3)(x + 2), marking clearly the points of intersection with the axes.
b Hence sketch y
= (x + 2)2(x β 1)(x + 4).
11 a Sketch the gra
ph of y = x3 + 4x2 + 4x. (6 marks)
b The point with coordinates (
β1, 0) lies on the curve with
equation y = (x ... | [
-0.05483977869153023,
0.01898946799337864,
-0.0563930906355381,
-0.02154814451932907,
-0.040815871208906174,
0.03966899961233139,
-0.008281194604933262,
0.006534191779792309,
-0.06808431446552277,
0.01998276822268963,
0.020941298454999924,
-0.08860451728105545,
-0.020887719467282295,
-0.01... |
76
Chapter 4
Example 12
Given that f(x) = 9 β x2, sketch the curves with equations:
a y =
f(2x) b y =
2f(x)
a f(x) = 9 β x2
So f(x) = (3 β x)(3 + x )
The curve is y =
(3 β x )(3 + x )
0 = (3 β x )(3 + x )
So x = 3 or x = β 3
So the curve crosses the x -axis at (3, 0)
and (β3, 0).
When x = 0, y = 3 Γ 3 = 9
So th... | [
0.011153441853821278,
0.11540050804615021,
-0.06174931675195694,
-0.02507524937391281,
0.025157272815704346,
0.04966996982693672,
-0.04062652215361595,
0.012362489476799965,
-0.08057485520839691,
0.022153962403535843,
0.016351742669939995,
-0.039625946432352066,
0.013299515470862389,
0.010... |
77Graphs and transformations
Example 13
a Sketch the curve with equation y = x(x β 2)(x + 1).
b On the same axes, sk
etch the curves y = 2x(2x β 2)(2x + 1) and y = βx(x β 2)(x + 1).
a
Oy
x 2y = x(x β 2)( x + 1)
β1
b
y = x(x β 2)( x + 1)y = 2 x(2x β 2)(2 x + 1)y = βx(x β 2)( x + 1)
Oy
x 2 β1y = βx(x β 2)(x + 1) is a st... | [
-0.08607317507266998,
-0.013573684729635715,
-0.030756376683712006,
-0.07673896849155426,
-0.03592035174369812,
0.05368461459875107,
-0.055394891649484634,
-0.008264830335974693,
-0.07694945484399796,
-0.04549087956547737,
-0.005500972270965576,
-0.025399621576070786,
0.01709596998989582,
... |
78
Chapter 4
1 Apply the f
ollowing transformations to the curves with equations y = f(x) where:
i f(x
) = x2 ii f(x) = x3 iii f(x) = 1 __ x
In each case show both f(x
) and the transformation on the same diagram.
a f(2x) b f(βx
) c f( 1 _ 2 x) d f(4x) e f( 1 _ 4 x)
f 2f(x
) g βf(x
) h 4f(x
) ... | [
-0.008617733605206013,
0.025739924982190132,
-0.10825558006763458,
-0.049027103930711746,
0.015980763360857964,
0.08586245775222778,
0.014097458682954311,
0.004709181841462851,
-0.08724911510944366,
-0.029178299009799957,
0.017304832115769386,
-0.06409555673599243,
0.011816606856882572,
0.... |
79Graphs and transformations
4.7 Transforming functions
You can apply transformations to unfamiliar functions by considering how specific points and features
are transformed.
Example 15
The following diagram shows a sketch of the curve f(x) which passes through the origin. The points A(1, 4) and B(3, 1) also lie on ... | [
-0.016790170222520828,
0.02158244140446186,
-0.05481753498315811,
-0.07096415013074875,
-0.04640055075287819,
0.04527558386325836,
-0.01431749016046524,
0.0012071575038135052,
-0.011572487652301788,
-0.03747863322496414,
0.0028491325210779905,
-0.015574891120195389,
-0.027858257293701172,
... |
80
Chapter 4
d 2y = f(x) so y = 1 __ 2 f(x)
(3, )(1, 2)
1
2y = f(x)1
2y
x O
e y β 1 = f( x) so y = f( x) + 1
1y
x O(3, 2)(1, 5)
y = f( x) + 1Rearrange in the form y = β¦
Stretch f(x) by scale factor 1 _ 2 in the y-dir ection.
Rearrange in the form y = β¦
Translate f(x) 1 unit in the direction of the
p... | [
-0.047331757843494415,
0.03014230914413929,
-0.01388527825474739,
0.003610537853091955,
-0.005678805988281965,
0.09377336502075195,
0.06767749786376953,
-0.0213429294526577,
0.005796169862151146,
0.005806676112115383,
0.008952684700489044,
-0.013110651634633541,
0.015286065638065338,
0.057... |
81Graphs and transformations
3 The curve with equation
y = f(x) passes through the y
x OB
CD
A (β4, β6)β2 4
β3
points A(β4, β6), B(β2, 0), C(0, β3) and D(4, 0)
as shown in the diagram.
Sketch the following and give the coordinates of
the points A, B, C and D after each transformation.
a f(x
β 2) b f(x
) + 6 c... | [
-0.011856946162879467,
0.02744477614760399,
-0.042537104338407516,
-0.04090229421854019,
-0.01198847871273756,
0.06760215014219284,
-0.014975075609982014,
-0.05927043408155441,
-0.0823395624756813,
-0.008436844684183598,
-0.006486008875072002,
-0.04300575330853462,
-0.0010384637862443924,
... |
82
Chapter 4
1 a On the same axes sketch the gr
aphs of y = x2(x β 2) and y = 2x β x2.
b By solving a suitable equa
tion find the points of intersection of the two graphs.
2 a On the same axes sketch the curv
es with equations y = 6 __ x and y = 1 + x.
b The curves intersect at the points
A and B . Find th... | [
-0.01696573570370674,
0.07196259498596191,
-0.022713154554367065,
0.024965256452560425,
-0.01977226324379444,
0.07208871096372604,
0.008239323273301125,
0.01169667113572359,
-0.07628785818815231,
-0.006508714985102415,
0.00936982873827219,
-0.03800100088119507,
-0.012277021072804928,
0.013... |
83Graphs and transformations
7 The diagram sho
ws the graph of the quadratic function f(x).
x O13
(2, β1)y
y = f(x)
The graph meets the x-axis at (1, 0) and (3, 0) and the
minimum point is (2, β1).
a Find the equation of the gr
aph in the form
y = ax2 + bx = c (2 marks)
b On separate ax
es, sketch the graphs o... | [
0.00649389810860157,
0.0698540061712265,
-0.018366096541285515,
-0.10235852748155594,
-0.03687293082475662,
0.06967819482088089,
-0.022405508905649185,
-0.019884124398231506,
-0.05680349841713905,
-0.04473491758108139,
0.000046465018385788426,
-0.02134426310658455,
-0.023704620078206062,
-... |
84
Chapter 4
13 Given tha
t f(x) = 1 __ x , x β 0,
a Sketch the gra
ph of y = f(x) β 2 and state the equations of the asymptotes. (3 marks)
b Find the coordinates of
the point where the curve y = f(x) β 2 cuts a coordinate
axis. (2 marks)
c Sketch the gra
ph of y = f(x + 3). (2 marks)
d State the equations o... | [
0.005548852030187845,
0.11908348649740219,
0.001617048867046833,
0.014689534902572632,
-0.009964574128389359,
-0.019769061356782913,
0.0358993262052536,
0.10491347312927246,
-0.018443966284394264,
0.0034839999862015247,
0.07910178601741791,
-0.06592093408107758,
-0.05153549462556839,
0.013... |
Review exercise
851
1 a Write down the value of 8 1 _ 3 . (1 mark)
b Find the value of
8 β 2 _ 3 . (2 marks)
β Section 1.4
2 a Find the value of 12 5 4 _ 3 . (2 marks)
b Simplify 24x2 Γ· 18 x 4 _ 3 . (2 marks)
β Sections 1.1, 1.4
3 a Express ββ―___ 80 in the form a ββ―__
5 , ... | [
0.0707051083445549,
0.08695950359106064,
-0.004012358840554953,
-0.05049767345190048,
0.01238033827394247,
0.12074267119169235,
-0.012086556293070316,
-0.00391438277438283,
-0.0718291699886322,
-0.0012865080498158932,
0.01211061142385006,
-0.09725109487771988,
0.027754759415984154,
-0.0240... |
86
Review exercise 1
11 The functions f and g are defined as
f(
x) = x(x β 2) and g(x) = x + 5, x β β .
Given tha
t f(a) = g(a) and a > 0,
find the value of a to three significant figures.
(3 marks)
β Sections 2.1, 2.3
12 An athlete launches a shot put from shoulder height. The height of the shot put, in metr
es... | [
0.015314262360334396,
0.11579277366399765,
0.06447824835777283,
-0.059130195528268814,
-0.035781342536211014,
0.02937139943242073,
0.024608595296740532,
0.03117590956389904,
-0.031325217336416245,
0.03970177844166756,
0.059722110629081726,
-0.03183305263519287,
-0.05115228146314621,
0.0443... |
87
Review exercise 1
18 a By eliminating
y from the equations:
y = x β 4,
2x2 β xy = 8,
show that
x2 + 4x β 8 = 0. (2 marks)
b Hence, or otherwise, solv
e the
simultaneous equations:
y = x β 4,2x
2 β xy = 8,
giving your answers in the form a Β± b
ββ―__
3 , where a and b are
integers. (4 marks)
β Section 3.2
... | [
0.008966542780399323,
0.07969505339860916,
0.008859998546540737,
-0.0068662879057228565,
0.007101622875779867,
0.012006348930299282,
0.012291046790778637,
-0.02132152020931244,
-0.06928626447916031,
0.03507368639111519,
0.08422739803791046,
-0.0928497388958931,
0.06862367689609528,
-0.0048... |
88
Review exercise 1
27
13
4
Oy
x
The figure shows a sketch of the curve
with equation y = f(x). The curve passes through the points (0, 3) and (4, 0) and touches the x-axis at the point (1, 0).
On separate diagrams, sketch the curves
with equations
a
y = f(
x + 1) (2 marks)
b y =
2f(x) (2 marks)
c y = f ( 1... | [
0.03879060968756676,
0.057683419436216354,
-0.027005931362509727,
0.0028135222382843494,
0.012351617217063904,
0.0805431380867958,
0.003618121612817049,
0.01488406676799059,
-0.04734433814883232,
0.018454739823937416,
0.04179245978593826,
-0.0864107683300972,
-0.024212485179305077,
0.04946... |
89
Straight line graphs
After completing this unit you should be able to:
β Calculat
e the gradient of a line joining a pair of points β pages 90 β 91
β Understand the link between the equation o f a line, and its gradient
and intercept β pages 91 β 93
β Find the equation of a line given (i) the gr adient and one po... | [
0.024515662342309952,
0.05557006597518921,
0.0007674279040656984,
-0.04259038716554642,
-0.022908538579940796,
0.04945764318108559,
-0.05691401660442352,
-0.05434376373887062,
-0.10876592248678207,
0.01933881640434265,
0.0350445993244648,
-0.0353688970208168,
-0.008358057588338852,
0.00601... |
90
Chapter 5
y
x O(x2, y2)
(x1, y1)x2 β x 1y2 β y 15.1 y = mx + c
You can find the gradient of a straight line joining two points
by considering the vertical distance and the horizontal distance between the points.
β The gradient m of a line joining the point with coordinates
( x 1 , y 1 ) to the point with c... | [
-0.01725941151380539,
-0.0033452033530920744,
0.028358004987239838,
0.00020366208627820015,
-0.0051992908120155334,
0.08801115304231644,
0.021469373255968094,
-0.03211066499352455,
-0.0009195214370265603,
0.03231723606586456,
0.0053368001244962215,
-0.010094579309225082,
0.08013003319501877,... |
91Straight line graphs
2 The line joining (3, β5) to (6,
a) has a gradient 4. Work out the value of a.
3 The line joining (5, b) to (8, 3) has gr
adient β3. Work out the value of b.
4 The line joining (c, 4) to (7, 6) has gr
adient 3 _ 4 . Work out the value of c.
5 The line joining (β1, 2
d ) to (1, 4) has gradi... | [
0.028725745156407356,
0.03366002440452576,
0.0209120512008667,
-0.06412093341350555,
-0.03791247680783272,
0.03455302119255066,
-0.01980348862707615,
-0.07750751078128815,
-0.08368227630853653,
0.023953501135110855,
0.01275207195430994,
-0.03535762056708336,
-0.02082219533622265,
-0.013173... |
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