1. Graph Transformations
Higher Unit 1
What is a set
Recognising a Function in various formats
Composite Functions
Exponential and Log Graphs
Graph Transformations
Trig Graphs
Connection between Radians and degrees & Exact values
Solving Trig Equations
Basic Trig Identities
Exam Type Questions

2. Sets & Functions Notation & Terminology
SETS: A set is a collection of items which have some common property.
These items are called the members or elements of the set.
Sets can be described or listed using “curly bracket” notation.

3. Sets & Functions eg {colours in traffic lights} = {red, amber, green}
DESCRIPTION
LIST
eg {square nos. less than 30}
= { 0, 1, 4, 9, 16, 25}
NB: Each of the above sets is finite because we can list every member

4. We can describe numbers by the following sets:
Sets & Functions
We can describe numbers by the following sets:
N = {natural numbers}
= {1, 2, 3, 4, ……….}
W = {whole numbers}
= {0, 1, 2, 3, ………..}
Z = {integers}
= {….-2, -1, 0, 1, 2, …..}
Q = {rational numbers}
This is the set of all numbers which can be written as fractions or ratios.
eg 5 = 5/ = -7/ = 6/10 = 3/5
55% = 55/100 = 11/20 etc

5. We should also note that
Sets & Functions
R = {real numbers}
This is all possible numbers. If we plotted values on a number line then each of the previous sets would leave gaps but the set of real numbers would give us a solid line.
We should also note that
N “fits inside” W
W “fits inside” Z
Z “fits inside” Q
Q “fits inside” R

6. Sets & Functions N W Z Q R When one set can fit inside another we say
that it is a subset of the other.
The members of R which are not inside Q are called irrational numbers. These cannot be expressed as fractions and include  , 2, 35 etc

7. This set has no elements and is called the empty set.
Sets & Functions
To show that a particular element/number belongs to a particular set we use the symbol .
eg 3  W but  Z
Examples
{ x  W: x < 5 }
= { 0, 1, 2, 3, 4 }
{ x  Z: x  -6 }
= { -6, -5, -4, -3, -2, …….. }
{ x  R: x2 = -4 }
= { } or 
This set has no elements and is called the empty set.

8. Functions & Mappings
Defn: A function or mapping is a relationship between two sets in which each member of the first set is connected to exactly one member in the second set.
If the first set is A and the second B then we often write f: A  B
The members of set A are usually referred to as the domain of the function (basically the starting values or even x-values) while the corresponding values or images come from set B and are called the range of the function (these are like y-values).

9. Functions & Mapping Example Suppose that f: A  B is defined by
Functions can be illustrated in three ways:
1) by a formula.
2) by arrow diagram.
3) by a graph (ie co-ordinate diagram).
Example
Suppose that f: A  B is defined by
f(x) = x2 + 3x where A = { -3, -2, -1, 0, 1}.
FORMULA
then f(-3) = 0 , f(-2) = -2 , f(-1) = -2 , f(0) = 0 , f(1) = 4
NB: B = {-2, 0, 4} = the range!

10. Functions & Mapping f(x) ARROW DIAGRAM A B f(-3) = 0 f(-2) = -2-2 -2 -1 -2 1 4

11. NB: This graph consists of 5 separate points. It is not a solid curve.
Functions & Graphs
In a GRAPH we get :
NB: This graph consists of 5 separate points. It is not a solid curve.

12. Recognising Functions
Functions & Graphs
Recognising Functions
A B
a b c d e f g
A B e f g
a bc d
YES
Not a function
two arrows leaving b!

13. Not a function - d unused!
Functions & Graphs
A B
Not a function - d unused!
a b c d
e f g
A B
a bc d
e f g h
YES

14. Recognising Functions from Graphs
Functions & Graphs
Recognising Functions from Graphs
If we have a function f: R  R (R - real nos.) then every vertical line we could draw would cut the graph exactly once!
This basically means that every x-value has one, and only one, corresponding y-value!

15. Function & Graphs Y
Function !! x

16. Function & Graphs Y Not a function !! Cuts graph more than once !
x must map to
one value of y x

17. Functions & Graphs
Not a function !! Y
Cuts graph
more than once! X

18. Functions & Graphs Y
Function !! X

19. COMPOSITION OF FUNCTIONS ( or functions of functions )
Composite Functions
COMPOSITION OF FUNCTIONS
( or functions of functions )
Suppose that f and g are functions where
f:A  B and g:B  C
with f(x) = y and g(y) = z
where x A, y B and z C.
Suppose that h is a third function where
h:A  C with h(x) = z .

20. We can say that h(x) = g(f(x)) “function of a function”
Composite Functions ie
A B C g f x y z h
We can say that h(x) = g(f(x))
“function of a function”

21. Suppose that f(x) = 3x - 2 and g(x) = x2 +1
Composite Functions
g(2)= =5
f(5)=5x3-2 =13
Example 1
f(1)=3x1 - 2 =1
f(1)=3x1 - 2 =1
Suppose that f(x) = 3x and g(x) = x2 +1
(a) g( f(2) ) =
g(4)
= 17
g(26)= =677
g(5)= =26
(b) f( g (2) ) =
f(5)
= 13
(c) f( f(1) ) =
f(1)
= 1
(d) g( g(5) )
= g(26)
= 677

22. Composite Functions Suppose that f(x) = 3x - 2 and g(x) = x2 +1
Find formulae for (a) g(f(x)) (b) f(g(x)).
(a) g(f(x)) = g(3x-2)
= (3x-2)2 + 1
= 9x2 - 12x + 5
(b) f(g(x)) = f(x2 + 1)
= 3(x2 + 1) - 2
= 3x2 + 1
NB: g(f(x))  f(g(x)) in general.
CHECK
g(f(2)) =
9 x x 2 + 5 =
= 17
f(g(2)) =
3 x
= 13

23. Composite Functions
Let h(x) = x - 3 , g(x) = x and k(x) = g(h(x)).
If k(x) = 8 then find the value(s) of x.
k(x) = g(h(x))
Put x2 - 6x = 8
= g(x - 3)
then x2 - 6x + 5 = 0
= (x - 3)2 + 4
or (x - 5)(x - 1) = 0
= x2 - 6x + 13
So x = 1 or x = 5
CHECK
g(h(5))
= g(2) =
= 8

24. Composite Functions Choosing a Suitable Domain
(i) Suppose f(x) = x2 - 4
Clearly x  0
So x2  4
So x  -2 or 2
Hence domain = {xR: x  -2 or 2 }

25. Composite Functions x = 0 (0 + 4)(0 - 2) = negative
(ii) Suppose that g(x) = (x2 + 2x - 8)
x = 3
(3 + 4)(3 - 2) = positive
We need (x2 + 2x - 8)  0
x = -5
(-5 + 4)(-5 - 2) = positive
Suppose (x2 + 2x - 8) = 0 -4 2
Then (x + 4)(x - 2) = 0
So x = -4 or x = 2
Check values below -4 , between -4 and 2, then above 2
So domain = { xR: x  -4 or x  2 }

26. Exponential (to the power of) Graphs
Exponential Functions
A function in the form f(x) = ax where a > 0, a ≠ 1
is called an exponential function to base a .
Consider f(x) = 2x x
f(x)
1 1/8 ¼ ½

27. Graph The graph of y = 2x (1,2) (0,1) Major Points
(i) y = 2x passes through the points (0,1) & (1,2)
(ii) As x ∞ y ∞ however as x -∞ y 0 .
(iii) The graph shows a GROWTH function.

28. Log Graphs y = log2x ie x 1/8 ¼ ½ 1 2 4 8 y -3 -2 -1 0 1 2 3
To obtain y from x we must ask the question
“What power of 2 gives us…?”
This is not practical to write in a formula so we say
“the logarithm to base 2 of x”
y = log2x
or “log base 2 of x”

29. Graph y = log2x (2,1) (1,0) The graph of NB: x > 0 Major Points
(i) y = log2x passes through the points (1,0) & (2,1) .
As x ∞ y ∞ but at a very slow rate and as x  0 y  -∞ .

30. y = ax Exponential (to the power of) Graphs
The graph of y = ax always passes through (0,1) & (1,a)
It looks like .. Y
y = ax
(1,a)
(0,1) x

31. The graph of y = logax always passes through (1,0) & (a,1)
Log Graphs
The graph of y = logax always passes through (1,0) & (a,1) Y
It looks like ..
(a,1)
(1,0) x
y = logax

32. Graph Transformations
We will investigate f(x) graphs of the form
1. f(x) ± k
f(x ± k)
-f(x)
f(-x)
kf(x)
f(kx)
Each moves the
Graph of f(x) in a certain way !

33. Graph of f(x) ± k Transformations3
y = f(x)
y = x2 2
y = f(x) ± k
y = x2-3 1
y = x2+ 1
Mathematically
y = f(x) ± k
moves f(x) up or down
Depending on the value of k
+ k move up
- k move down -2 -1 1 2 -1 -2 -3

34. Graph of -f(x ± k) Transformations
Mathematically
y = f(x ± k)
moves f(x) to the left or right
depending on the value of k
-k move right
+ k move left
Graph of -f(x ± k) Transformations
y = f(x)
y = x2
y = f(x ± k)
y = (x-1)2
y = (x+2)2 3 2 1 -4 -3 -2 -1 1 2 3

35. Graph of -f(x) Transformations
y = f(x)
y = x2
y = -f(x)
y = -x2
Mathematically
y = –f(x)
reflected f(x) in the x - axis

36. Graph of -f(x) Transformations
y = f(x)
y = 2x + 3
y = -f(x)
y = -(2x + 3)
Mathematically
y = –f(x)
reflected f(x) in the x - axis

37. Graph of -f(x) Transformations
y = f(x)
y = x3
y = -f(x)
y = -x3
Mathematically
y = –f(x)
reflected f(x) in the x - axis

38. Graph of f(-x) Transformations
y = f(x)
y = x + 2
y = f(-x)
y = -x + 2
Mathematically
y = f(-x)
reflected f(x) in the y - axis

39. Graph of f(-x) Transformations
y = f(x)
y = (x+2)2
y = f(-x)
y = (-x+2)2
Mathematically
y = f(-x)
reflected f(x) in the y - axis

40. Graph of k f(x) Transformations
y = f(x)
y = x2-1 3
y = k f(x)
y = 4(x2-1) 2
y = 0.25(x2-1) 1
Mathematically
y = k f(x)
Multiply y coordinate by a factor of k
k > 1  (stretch in y-axis direction)
0 < k < 1  (squash in y-axis direction) -2 -1 1 2 -1 -4

41. Graph of -k f(x) Transformations
y = f(x)
y = x2-1
y = k f(x)
y = -4(x2-1) 4
y = -0.25(x2-1)
Mathematically
y = -k f(x)
k = -1 reflect graph in x-axis
k < -1  reflect f(x) in x-axis & multiply by a factor k (stretch in y-axis direction)
0 < k < -1  reflect f(x) in x-axis multiply by a factor k (squash in y-axis direction) 1 -2 -1 1 2 -1 -4

42. Graph of f(kx) Transformations4
y = f(x)
y = (x-2)2 3
y = f(kx)
y = (2x-2)2 2
y = (0.5x-2)2 1 1 2 3 4 5 6
Mathematically
y = f(kx)
Multiply x – coordinates by 1/k
k > 1  squashes by a factor of 1/k in the x-axis direction
k < 1  stretches by a factor of 1/k in the x-axis direction

43. Graph of f(-kx) Transformations4
y = f(x)
y = (x-2)2 3
y = f(-kx)
y = (-2x-2)2 2 1
y = (-0.5x-2)2 -4 -3 -2 -1 1 2 3 4
Mathematically
y = f(-kx)
k = -1 reflect in y-axis
k < -1  reflect & squashes by factor of 1/k in x direction
-1 < k > 0  reflect & stretches factor of 1/k in x direction

44. Trig Graphs The same transformation rules
apply to the basic trig graphs.
NB: If f(x) =sinx then 3f(x) = 3sinx
and f(5x) = sin5x
Think about sin replacing f !
Also if g(x) = cosx then g(x) – 4 = cosx  – 4
and g(x + 90) = cos(x + 90) 
Think about cos replacing g !

45. Trig Graphs y = sinx - 2 Sketch the graph of y = sinx - 2
If sinx = f(x) then sinx - 2 = f(x) - 2
So move the sinx graph 2 units down. 1
90o
180o
270o
360o -1 -2
y = sinx - 2 -3

46. Trig Graphs y = cos(x - 50)o Sketch the graph of y = cos(x - 50)
If cosx = f(x) then cos(x - 50) = f(x - 50)
So move the cosx graph 50 units right.
50o 1
90o
180o
270o
360o -1 -2
y = cos(x - 50)o -3

47. Trig Graphs y = 3sinx Sketch the graph of y = 3sinx
If sinx = f(x) then 3sinx = 3f(x)
So stretch the sinx graph 3 times vertically. 3 2 1
90o
180o
270o
360o -1 -2
y = 3sinx -3

48. Trig Graphs y = cos4x Sketch the graph of y = cos4x
If cosx = f(x) then cos4x = f(4x)
So squash the cosx graph to 1/4 size horizontally 1
90o
180o
270o
360o -1
y = cos4x

49. Trig Graphs y = 2sin3x Sketch the graph of y = 2sin3x
If sinx = f(x) then 2sin3x = 2f(3x)
So squash the sinx graph to 1/3 size horizontally and also double its height. 3 2 1
90o
180o
270o
360o -1 -2
y = 2sin3x -3

50. Radians
Radian measure is an alternative to degrees and is based upon the ratio of
arc length radius a θ
θ(radians) = r
θ- theta

51. Radians θ (radians) = r/r = 1 So θ (radians) = πr /r = π a
a = r
If the arc length = the radius θ r
θ (radians) = r/r = 1
If we now take a semi-circle a
Here a = ½ of circumference
= ½ of πd
θ (180o) r
= πr
So θ (radians) = πr /r = π

52. Converting
then X π
÷180
degrees
radians
÷ π
then x 180

53. Radians Since we have a semi-circle the angle must be 180o.
We now get a simple connection between degrees and radians.
π (radians) = 180o
This now gives us
2π = 360o
π /2 = 90o
3π /2 = 270o
π /3 = 60o
2π /3 = 120o
π /4 = 45o
3π /4 = 135o
π /6 = 30o
5π /6 = 150o
NB: Radians are usually expressed as fractional multiples of π.

54. Converting Ex1 72o = 72/180 X π = 2π /5 Ex2 330o = 330/180 X π =
11 π /6
Ex3 2π /9 =
2π /9 ÷ π x 180o =
2/9 X 180o =
40o
Ex4 23π/18 =
23π /18 ÷ π x 180o =
23/18 X 180o =
230o

55. This triangle will provide exact values for
Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values
60º 2 2
60º
30º 3 1
This triangle will provide exact values for
sin, cos and tan 30º and 60º

56. Exact Values ½  ½ 1 1 x 0º 30º 45º 60º 90º 3 Sin xº Cos xº Tan xº 32 1 3 2 1 3

57. Exact Values 2 45º 1 1 45º 1 1 Consider the square with sides 1 unit
We are now in a position to calculate
exact values for sin, cos and tan of 45o

58. Exact Values ½ ½ 1 1 1 x 0º 30º 45º 60º 90º 1 2 1 2 3 Sin xº Cos xº
Tan xº
Undefined
1 2 3 2 1 3 2
1 2 1 1 3

59. Exact value table and quadrant rules. tan150o = - tan(180 - 150) o
= -1/√3
(Q2 so neg)
cos300o
= cos( ) o
= cos60o
= 1/2
(Q4 so pos)
sin120o
= sin( ) o
= sin60o
= √ 3/2
(Q2 so pos)
tan300o
= - tan( )o
= - tan60o
= - √ 3
(Q4 so neg)

60. Find the exact value of cos2(5π/6) – sin2(π/6)
Exact value table
and quadrant rules.
Find the exact value of cos2(5π/6) – sin2(π/6)
cos(5π/6) =
cos150o
= cos( )o
= - cos30o
= - √3 /2
(Q2 so neg)
= 1/2
sin(π/6)
= sin30o
cos2(5π/6) – sin2(π/6)
= (- √3 /2)2 – (1/2)2
= ¾ - 1/4
= 1/2

61. Prove that sin(2 π /3) = tan (2 π /3)
Exact value table
and quadrant rules.
Prove that sin(2 π /3) = tan (2 π /3)
cos (2 π /3)
sin(2π/3) = sin120o
= sin(180 – 120)o
= sin60o
= √3/2
cos(2 π /3) = cos120o
= cos(180 – 120)o
= - cos60o
= -1/2
tan(2 π /3) = tan120o
= tan(180 – 120)o
= -tan60o
= - √3
sin(2 π /3) cos (2 π /3)
LHS =
= √3/2 ÷ -1/2
= √3/2 X -2
= - √3
= tan(2π/3)
= RHS

62. Solving Trig Equations1 2 3 4
Sin +ve
All +ve
180o - xo
180o + xo
360o - xo
Tan +ve
Cos +ve
created by Mr. Lafferty

63. Solving Trig EquationsC A S T 0o
180o
270o
90o
Solving Trig Equations
Graphically what are we trying to solve
Example 1 Type 1:
Solving the equation sin xo = 0.5 in the range 0o to 360o 1 2 3 4
sin xo = (0.5)
xo = sin-1(0.5)
xo = 30o
There is another solution
xo = 150o
(180o – 30o = 150o)
created by Mr. Lafferty

64. Solving Trig EquationsC A S T 0o
180o
270o
90o
Solving Trig Equations
Graphically what are we trying to solve
Example 2 :
Solving the equation cos xo = 0 in the range 0o to 360o 1 2 3 4
cos xo = 0.625
xo = cos -1 (0.625)
xo = 51.3o
There is another solution
xo = 308.7o
(360o o = 308.7o)
created by Mr. Lafferty

65. Solving Trig EquationsC A S T 0o
180o
270o
90o
Solving Trig Equations
Graphically what are we trying to solve
Example 3 :
Solving the equation tan xo – 2 = 0 in the range 0o to 360o 1 2 3 4
tan xo = 2
xo = tan -1(2)
xo = 63.4o
There is another solution
x = 180o o = 243.4o
created by Mr. Lafferty

66. Solving Trig EquationsC A S T 0o
180o
270o
90o
Solving Trig Equations
Graphically what are we trying to solve
Example 4 Type 2 :
Solving the equation sin 2xo = 0 in the range 0o to 360o
sin 2xo = (-0.6)
2xo = sin-1(0.6)
2xo = 37o ( always 1st Q First)
2xo = 217o , 323o
577o , 683o ÷2
xo = o , 161.5o
288.5o , 341.5o
created by Mr. Lafferty

67. Solving Trig EquationsC A S T 0o
180o
270o
90o
Solving Trig Equations
Graphically what are we trying to solve
Example 5 Type 3 :
Solving the equation 2sin (2xo - 30o) - √3 = 0 in the range 0o to 360o
2sin (2x - 30o) = √3
sin (2x - 30o) = √3 ÷ 2
2x - 30o = sin-1(√3 ÷ 2)
2xo - 30o = 60o , 120o ,420o , 480o
2xo = 90o , 150o ,450o , 530o ÷2
xo = 45o , 75o
225o , 265o
created by Mr. Lafferty

68. Solving Trig EquationsC A S T 0o
180o
270o
90o
Solving Trig Equations
Graphically what are we trying to solve
Example 5 Type 4 :
Solving the equation cos2x = 1 in the range 0o to 360o
cos2 xo = 1
cos xo = ± 1
cos xo = 1
xo = 0o and 360o
cos xo = -1
xo = 180o
created by Mr. Lafferty

69. Solving Trig EquationsC A S T 0o
180o
270o
90o C A S T 0o
180o
270o
90o
Solving Trig Equations
Example 6 Type 5 :
Solving the equation
3sin2x + 2sin x - 1 = 0 in the range 0o to 360o
Let p = sin x
We have 3p2 + 2p - 1 = 0
Factorise
(3p – 1)(p + 1) = 0
3p – 1 = 0
p + 1 = 0
p = 1/3
p = - 1
sin x = 1/3
sin x = -1
xo = 19.5o and 160.5o
xo = 270o

70. Trig Identities
An identity is a statement which is true for all values.
eg 3x(x + 4) = 3x2 + 12x
eg (a + b)(a – b) = a2 – b2
Trig Identities
(1) sin2θ + cos2 θ = 1
(2) sin θ = tan θ cos θ
θ ≠ an odd multiple of π/2 or 90°.

71. Trig Identities (1) sin2θo + cos2 θo = a2 +b2 = c2 c a sinθo = a/c b
Reason
a2 +b2 = c2 c a
sinθo = a/c θo b
cosθo = b/c
(1) sin2θo + cos2 θo =

72. Simply rearranging we get two other forms
Trig Identities
Simply rearranging we get two other forms
sin2θ + cos2 θ = 1
sin2 θ = 1 - cos2 θ
cos2 θ = 1 - sin2 θ

73. Since θ is between 0 < θ < π/2
Trig Identities
Example1
sin θ = 5/13 where 0 < θ < π/2
Find the exact values of cos θ and tan θ .
cos2 θ = 1 - sin2 θ
Since θ is between 0 < θ < π/2
then cos θ > 0
= 1 – (5/13)2
So cos θ = 12/13
= 1 – 25/169
tan θ = sinθ cos θ
= 5/13 ÷ 12/13
= 144/169
cos θ = √(144/169)
= 5/13 X 13/12
= 12/13 or -12/13
tan θ = 5/12

74. Trig Identities Given that cos θ = -2/ √ 5 where π< θ < 3 π /2
Find sin θ and tan θ.
Since θ is between π< θ < 3 π /2
sinθ < 0
sin2 θ = 1 - cos2 θ
= 1 – (-2/ √ 5 )2
Hence sinθ = - 1/√5
= 1 – 4/5
tan θ = sinθ cos θ
= / √ 5 ÷ -2/ √ 5
= 1/5
= / √ 5 X - √5 /2
sin θ = √(1/5)
= 1/ √ 5 or - 1/ √ 5
Hence tan θ = 1/2

75. Higher Maths
Graphs & Functions
Strategies
Click to start

76. Graphs & Functions Higher
The following questions are on
Graphs & Functons
Non-calculator questions will be indicated
You will need a pencil, paper, ruler and rubber.
Click to continue

77. Graphs & Functions Higher
The diagram shows the graph of a function f.
f has a minimum turning point at (0, -3) and a
point of inflexion at (-4, 2).
a) sketch the graph of y = f(-x).
b) On the same diagram, sketch the graph of y = 2f(-x) a)
Reflect across the y axis b)
Now scale by 2 in the y direction
Hint
Previous
Quit
Quit
Next

78. Graphs & Functions Higher
The diagram shows a sketch of part of
the graph of a trigonometric function
whose equation is of the form
Determine the values of a, b and c 2a 1
a = 4
1 in 
2 in 2 
a is the amplitude:
b = 2
b is the number of waves in 2
c = 1
c is where the wave is centred vertically
Hint
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79. Graphs & Functions Higher
Functions and are defined on suitable domains.
a) Find an expression for h(x) where h(x) = f(g(x)).
b) Write down any restrictions on the domain of h. a) b)
Hint
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80. Graphs & Functions Higher
a) Express in the form
b) On the same diagram sketch
i) the graph of
ii) the graph of
c) Find the range of values of x for
which is positive b) a) c)
Solve:
Hint
10 - f(x) is positive for < x < 5
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81. Graphs & Functions Higher
The graph of a function f intersects the x-axis at (–a, 0)
and (e, 0) as shown.
There is a point of inflexion at (0, b) and a maximum turning
point at (c, d).
Sketch the graph of the derived function
m is +
m is +
m is -
Hint
f(x)
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82. Graphs & Functions Higher
Functions f and g are defined on suitable domains by and
a) Find expressions for: i)
ii)
b) Solve a) b)
Hint
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83. Graphs & Functions Higher
The diagram shows the graphs of two quadratic
functions
Both graphs have a minimum turning point at (3, 2).
Sketch the graph of
and on the same diagram
sketch the graph of
y=f(x)
y=g(x)
Hint
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84. Graphs & Functions Higher
are defined on a suitable set of real numbers.
a) Find expressions for
b) i) Show that
ii) Find a similar expression for
and hence solve the equation a) b)
Now use exact values
Repeat for ii)
Hint
equation reduces to
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85. Graphs & Functions Higher
A sketch of the graph of y = f(x) where is shown.
The graph has a maximum at A and a minimum at B(3, 0)
a) Find the co-ordinates of the turning point at A.
b) Hence, sketch the graph of
Indicate the co-ordinates of the turning points. There is no need to
calculate the co-ordinates of the points of intersection with the axes.
c) Write down the range of values of k for which g(x) = k has 3 real roots. a)
Differentiate
for SP, f(x) = 0
when x = 1
t.p. at A is: b)
Graph is
moved 2 units to the left, and 4 units up
t.p.’s are:
Hint c)
For 3 real roots, line y = k has to cut graph at 3 points
from the graph, k  4
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86. Graphs & Functions Higher
a) Find
b) If find in its simplest form. a) b)
Hint
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87. Graphs & Functions Higher
Part of the graph of is shown in the diagram.
On separate diagrams sketch the graph of
a) b)
Indicate on each graph the images of O, A, B, C, and D. a)
graph moves to the left 1 unit b)
graph is reflected in the x axis
graph is then scaled 2 units in the y direction
Hint
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88. Graphs & Functions Higher
Functions f and g are defined on the set of real numbers by
a) Find formulae for
i) ii)
b) The function h is defined by
Show that and sketch the graph of h.
c) Find the area enclosed between this graph and the x-axis. a) b) c)
Graph cuts x axis at 0 and 1
Now evaluate
Hint
Area
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89. Graphs & Functions Higher
The functions f and g are defined on a suitable domain by
a) Find an expression for b) Factorise a) b)
Difference of 2 squares
Simplify
Hint
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90. Graphs & Functions Higher
You have completed all 13 questions in this section
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91. Graphs & Functions Higher
Table of exact values
30°
45°
60°
sin
cos
tan 1
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