Higher Unit 1

1
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
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
Sets Functions
We can describe numbers by the following sets
1, 2, 3, 4, .
N natural numbers
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 5/1
-7 -7/1 0.6 6/10 3/5
55 55/100 11/20 etc
5
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
Q
Z
W
N
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
Sets Functions
To show that a particular element/number belongs
to a particular set we use the symbol ?.
eg 3 ? W but 0.9 ? Z
Examples
x ? W x lt 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
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
ARROW DIAGRAM
A B
f(x)
-3
0
f(-3) 0 f(-2) -2 f(-1) -2 f(0) 0 f(1) 4
-2
-2
-1
-2
0
0
0
1
4
11
Functions Graphs
In a GRAPH we get
NB This graph consists of 5 separate points.
It is not a solid curve.
12
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
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
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
Composite Functions
COMPOSITION OF FUNCTIONS
( or functions of functions )
Suppose that f and g are functions where
fA ? B and gB ? 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 hA
? C with h(x) z .
20
Composite Functions
ie
A B C
g
f
y
z
x
h
We can say that h(x) g(f(x))
function of a function
21
g(4)42 1 17
f(2)3x2 2 4
Composite Functions
f(5)5x3-2 13
g(2)22 1 5
Example 1
f(1)3x1 - 2 1
f(1)3x1 - 2 1
Suppose that f(x) 3x - 2 and g(x) x2 1
(a) g( f(2) )
g(4)
17
g(26)262 1 677
g(5)52 1 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 22 - 12 x 2 5
36 - 24 5
17
f(g(2))
3 x 22 1
13
23
Composite Functions
Let h(x) x - 3 , g(x) x2 4 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 13 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
8
CHECK
g(h(5))
g(2)
22 4
24
Composite Functions
Choosing a Suitable Domain
(i) Suppose f(x) 1 . x2 - 4
Clearly x2 - 4 ? 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
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
Graphs Functions

Higher
a)
Difference of 2 squares
b)
Simplify
27
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)
28
Graphs Functions

Higher
a)
b)
29
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.
a)
b)
30
Exponential (to the power of) Graphs
Exponential Functions
A function in the form f(x) ax where
a gt 0, a ? 1 is called an exponential function
to base a .

Consider f(x) 2x
x -3 -2 -1 0 1 2 3
f(x)
1 1/8 ¼ ½ 1 2
4 8
31
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 ?8 y ?8 however as x ?-8 y ?0 .
(iii) The graph shows a GROWTH function.
32
Log Graphs
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
33
Graph
(2,1)
(1,0)
The graph of
y log2x
NB x gt 0
Major Points
(i) y log2x passes through the points (1,0)
(2,1) .

• As x ?8 y ?8 but at a very slow rate
  and as x ? 0 y ? -8 .

34
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
35
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
36
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 !
37
Graph of f(x) k Transformations
3
y f(x)
y x2
2
y x2-3
y f(x) k
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
0
1
2
-1
-2
-1
-2
-3
38
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
y f(x)
y x2
y (x-1)2
y f(x k)
y (x2)2
3
2
1
0
1
2
-1
-2
-3
-4
3
39
f(x)
f(x) - 2
B(1,0)
A(-1,0)
C(0,-1)
B(1,-2)
A(-1,-2)
C(0,-3)
40
A(45o,0.5)
f(x)
B(90o,0)
C(135o,-0.5)
Plot f(x) 1
41
f(x) 1
A(45o,1.5)
B(90o,1)
C(135o,0.5)
A(45o,0.5)
B(90o,0)
C(135o,-0.5)
42
Graph of -f(x) Transformations
y f(x)
y x2
y -x2
y -f(x)
Mathematically y f(x) reflected f(x) in the
x - axis
43
Graph of -f(x) Transformations
y f(x)
y 2x 3
y -(2x 3)
y -f(x)
Mathematically y f(x) reflected f(x) in the
x - axis
44
Graph of -f(x) Transformations
y f(x)
y x3
y -x3
y -f(x)
Mathematically y f(x) reflected f(x) in the
x - axis
45
f(x)
- f(x)
C(0,1)
B(1,0)
A(-1,0)
A(-1,0)
B(1,0)
C(0,-1)
46
- f(x)
C(135o,0.5)
A(45o,0.5)
B(90o,0)
A(45o,-0.5)
C(135o,-0.5)
47
Graph of f(-x) Transformations
y f(x)
y x 2
y -x 2
y f(-x)
Mathematically y f(-x) reflected f(x) in the
y - axis
48
Graph of f(-x) Transformations
y f(x)
y (x2)2
y (-x2)2
y f(-x)
Mathematically y f(-x) reflected f(x) in the
y - axis
49
f(x)
f(-x)
C(0,1)
B(1,0)
A(1,0)
A(-1,0)
B(-1,0)
C(0,-1)
50
f(-x)
C (1,1)
A(-1,1)
B(0,0)
C(1,-1)
A(-1,-1)
51
Graph of k f(x) Transformations
y f(x)
y x2-1
3
y 4(x2-1)
y k f(x)
2
y 0.25(x2-1)
1
Mathematically y k f(x) Multiply y
coordinate by a factor of k k gt 1 ? (stretch in
y-axis direction) 0 lt k lt 1 ? (squash in y-axis
direction)
0
1
2
-1
-2
-1
-4
52
Graph of -k f(x) Transformations
y f(x)
y x2-1
y -4(x2-1)
y -k f(x)
4
y -0.25(x2-1)
Mathematically y -k f(x) k -1 reflect
graph in x-axis k lt -1 ? reflect f(x) in x-axis
multiply by a factor k (stretch in y-axis
direction) 0 lt k lt -1 ? reflect f(x) in x-axis
multiply by a factor k (squash in y-axis
direction)
1
0
1
2
-1
-2
-1
-4
53
Graph of f(kx) Transformations
4
y f(x)
y (x-2)2
3
y (2x-2)2
y f(kx)
2
y (2(x-1))2
1
y (0.5x-2)2
(0.5(x-4))2
0
1
2
4
6
5
3
Mathematically y f(kx) Multiply x
coordinates by 1/k k gt 1 ? squashes by a factor
of 1/k in the x-axis direction k lt 1 ? stretches
by a factor of 1/k in the x-axis direction
54
Graph of f(-kx) Transformations
4
y f(x)
y (x-2)2
3
2
y f(-kx)
y (-2x-2)2
1
y (-2(x1))2
-2
-1
1
3
2
0
4
-3
-4
y (-0.5x - 2)2
(-0.5(x 4))2
Mathematically y f(-kx) k -1 reflect in
y-axis k lt -1 ? reflect squashes by factor of
1/k in x direction -1 lt k gt 0 ? reflect
stretches factor of 1/k in x direction
55

• Explain the effect the following have
• -f(x)
• f(-x)
• f(x) k

(1,3)
(1,3)
(-1,-3)
(-1,-3)
2f(x) 1
f(x 1) 2
(1,3)
(1,3)
Name
f(-x) 1
(-1,-3)
-f(x) - 2
(-1,-3)
(1,3)

• Explain the effect the following have
• f(x k)
• kf(x)
• f(kx)

(1,3)
f(0.5x) - 1
-f(x 1) - 3
(-1,-3)
(-1,-3)
56

• Explain the effect the following have
• -f(x) flip in x-axis
• f(-x) flip in y-axis
• f(x) k move up or down

(1,7)
(0,5)
(1,3)
(1,3)
(-2,-1)
(-1,-3)
(-1,-3)
2f(x) 1
(-1,-5)
f(x 1) 2

(1,3)
(-1,4)

(1,3)
Name
(-1,1)

f(-x) 1
(-1,-3)
-f(x) - 2
(1,-2)
(1,-5)
(-1,-3)
(1,3)

• Explain the effect the following have
• f(x k) move left or right
• kf(x) stretch / compress
• in y direction
• f(kx) stretch / compress
• in x direction

(1,3)
f(0.5x) - 1
(-2,0)
(2,2)
-f(x 1) - 3
(-1,-3)
(-1,-3)

(-2,-4)
(0,-6)

57
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
58
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
graph is reflected in the x axis
b)
graph is then scaled 2 units in the y direction
59
Graphs Functions

Higher

a) On the same
diagram sketch i) the graph of
ii) the graph of b) Find the range of values
of x for which is
positive
a)

b)
Solve
10 - f(x) is positive for -1 lt x lt 5
60
(-1,8)
Graphs Functions

Higher
(1,4)


moved 2 units to the left, and 4 units up
Graph is
t.p.s are
61
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 !
62
Trig Graphs
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
0
90o
180o
270o
360o
-1
-2
y sinx? - 2
-3
63
Trig Graphs
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
0
90o
180o
270o
360o
-1
-2
y cos(x? - 50)o
-3
64
Trig Graphs
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
0
90o
180o
270o
360o
-1
-2
y 3sinx?
-3
65
Trig Graphs
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
0
90o
180o
270o
360o
-1
y cos4x?
66
Trig Graphs
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
0
90o
360o
180o
270o
-1
-2
y 2sin3x?
-3
67
Write down equations for graphs shown ?
y 0.5sin2xo 0.5 y 2sin4xo- 1
Write down the equations in the form f(x) for the
graphs shown?
y 0.5f(2x) 0.5 y 2f(4x) - 1
Trig Graph
Combinations
Higher
3
2
1
0
www.mathsrevision.com
90o
180o
270o
360o
-1
-2
-3
68
Write down the equations for the graphs shown?
Write down the equations in the form f(x) for the
graphs shown?
Trig Graphs
y cos2xo 1 y -2cos2xo - 1
y f(2x) 1 y -2f(2x) - 1
Combinations
Higher
3
2
1
0
www.mathsrevision.com
90o
180o
270o
360o
-1
-2
-3
69
Radians
Radian measure is an alternative to degrees and
is based upon the ratio of
arc Length radius
L
?
r
?- theta (angle at the centre)
So, full circle 360o ?2p radians
70
Radians
360o ? 2p
180o ? p
Copy Table
90o ?
270o ?
60o ?
120o ?
240o ?
300o ?
45o ?
135o ?
225o ?
315o ?
30o ?
150o ?
210o ?
330o ?
71
Converting
For any values
then X p
180
degrees
radians
p
then x 180
72
Converting
Ex1 72o
72/180 X p
2p /5
Ex2 330o
330/180 X p
11 p /6
Ex3 2p /9
2p /9 p x 180o
2/9 X 180o
40o
Ex4 23p/18
23p /18 p x 180o
23/18 X 180o
230o
73
Exact Values
Some special values of Sin, Cos and Tan are
useful left as fractions, We call these exact
values
30º
?3
1
This triangle will provide exact values for sin,
cos and tan 30º and 60º
74
Exact Values
?3 2
½
1
0
?3 2
1
½
0
0
?3
75
Exact Values
Consider the square with sides 1 unit
?2
45º
1
1
45º
1
1
We are now in a position to calculate exact
values for sin, cos and tan of 45o
76
Exact Values
?3 2
1 ?2
½
1
0
?3 2
1 ?2
1
½
0
0
1
?3
77
Exact value table and quadrant rules.
tan150o
- tan(180 - 150) o
- tan30o
-1/v3
(Q2 so neg)
cos300o
cos(360 - 300) o
cos60o
1/2
(Q4 so pos)
sin120o
sin(180 - 120) o
sin60o
v 3/2
(Q2 so pos)
tan300o
- tan(360-300)o
- tan60o
- v 3
(Q4 so neg)
78
Exact value table and quadrant rules.
Find the exact value of cos2(5p/6)
sin2(p/6)
cos(5p/6)
cos150o
cos(180 - 150)o
- cos30o
- v3 /2
(Q2 so neg)
1/2
sin(p/6)
sin30o
cos2(5p/6) sin2(p/6)
(- v3 /2)2 (1/2)2
¾ - 1/4
1/2
79
Exact value table and quadrant rules.
Prove that sin(2 p /3) tan (2 p /3) cos (2
p /3)
sin(2p/3) sin120o
sin(180 120)o
sin60o
v3/2
cos(2 p /3) cos120o
cos(180 120)o
- cos60o
-1/2
tan(2 p /3) tan120o
tan(180 120)o
-tan60o
- v3
sin(2 p /3) cos (2 p /3)
LHS
v3/2 -1/2
v3/2 X -2
- v3
tan(2p/3)
RHS
80
Solving Trig Equations
All ve
Sin ve
180o - xo
180o xo
360o - xo
Cos ve
Tan ve
81
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
sin xo (0.5)
xo sin-1(0.5)
xo 30o
There is another solution
xo 150o
(180o 30o 150o)
82
Solving Trig Equations
Graphically what are we trying to solve
Example 2 Solving the equation cos xo - 0.625
0 in the range 0o to 360o
cos xo 0.625
xo cos -1 (0.625)
xo 51.3o
There is another solution
xo 308.7o
(360o - 53.1o 308.7o)
83
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
tan xo 2
xo tan -1(2)
xo 63.4o
There is another solution
x 180o 63.4o 243.4o
84
Solving Trig Equations
Graphically what are we trying to solve
Example 4 Type 2 Solving the equation sin 2xo
0.6 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 108.5o , 161.5o 288.5o , 341.5o
85
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
86
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
(3p 1)(p 1) 0
Factorise
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
87
Solving Trig Equations
Graphically what are we trying to solve
Example 5 Type 3 Solving the equation 2sin
(2xo - 30o) - v3 0 in the range 0o to 360o
2sin (2x - 30o) v3
sin (2x - 30o) v3 2
2x - 30o sin-1(v3 2)
2xo - 30o 60o , 120o ,420o , 480o .........
2xo 90o , 150o ,450o , 510o .........
2
xo 45o , 75o 225o , 255o
88
Graphs Functions

Higher
Functions f and g are defined on suitable
domains by and a) Find
expressions for i) ii) b) Solve
a)
b)
89
Graphs Functions

Higher
a)
b)
Now use exact values
Repeat for ii)
equation reduces to
90
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
a 4
a is the amplitude
b 2
b is the number of waves in 2?
c 1
c is where the wave is centred vertically
91
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) sin2? cos2 ? 1
(2) sin ? tan ? cos ?
? ? an odd multiple of p/2 or 90.
92
Trig Identities
Reason
a2 b2 c2
c
a
sin?o a/c
?o
b
cos?o b/c
(1) sin2?o cos2 ?o
93
Trig Identities
Simply rearranging we get two other forms
sin2? cos2 ? 1
sin2 ? 1 - cos2 ?
cos2 ? 1 - sin2 ?
94
Trig Identities
Example1
sin ? 5/13 where 0 lt ? lt p/2
Find the exact values of cos ? and tan ? .
cos2 ? 1 - sin2 ?
Since ? is between 0 lt ? lt p/2 then cos ? gt 0
1 (5/13)2
So cos ? 12/13
1 25/169
tan ? sin? cos ?
5/13 12/13
144/169
cos ? v(144/169)
5/13 X 13/12
12/13 or -12/13
tan ? 5/12
95
Trig Identities
Given that cos ? -2/ v 5 where plt ? lt
3 p /2
Find sin ? and tan ?.
Since ? is between plt ? lt 3 p /2 sin? lt 0
sin2 ? 1 - cos2 ?
1 (-2/ v 5 )2
Hence sin? - 1/v5
1 4/5
tan ? sin? cos ?
- 1/ v 5 -2/ v 5
1/5
- 1/ v 5 X - v5 /2
sin ? v(1/5)
1/ v 5 or - 1/ v 5
Hence tan ? 1/2