Basic Skills in Higher Mathematics

1
Basic Skills in Higher Mathematics
Mathematics 1(H)Outcome 2
Robert GlenAdviser in Mathematics
2
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
y
Functions and graphs
x
3
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
y
PC Index
Click on the one you want
PC(a) Related functions and graphs
x
PC(b) Exponential and logarithmic graphs
PC(c) Composite functions
4
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Related functions and graphs
5
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Index Click on the one you want
1 y f (x) ? a Slide up/down
2 y f (x ? a) Slide left/right
3 y -f (x) Reflect in x-axis
4 y f (-x) Reflect in y-axis
5 y kf (x) Stretch/compress vertically

6 y f (kx) Stretch/compress
horizontally
6
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Section 1
y f (x) ? a Slide up/down
7
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
?
In this graph every point on y f(x) has moved
up 5 units.
y
y f (x) 5
?
5
?
?
y f (x)
5
5
5
-2
?
In this graph every point on y f(x) has moved
down 2 units.
y f (x) - 2
?
?
-2
-2
?
0
?
x
-2
8
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
In this graph every point on y f(x) has moved
up a units.
y
y f (x) a
y f (x)
In this graph every point on y f(x) has moved
down b units.
y f (x) - b
a
x
-b
9
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
1 Write down an equation for each graph.
y
a
y x2
3
Here are the answers
b
0
x
a
y x2 3
-1
b
y x2 - 1
10
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
2 Write down an equation for each graph.
y
a
9
y x3 - 5x
0
x
Here are the answers
a
y x3 - 5x 9
b
-8
b
y x3 - 5x - 8
11
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
3 Write down an equation for each graph.
y
a
8
6
4
y f (x)
2
0
x
Here are the answers
-2
b
-4
a
y f (x) 4
-6
b
y f (x) - 6
12
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Section 2
y f (x ? a) Slide left/right
13
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
y
In this graph every point on y f(x) has moved 5
units to the left.
In this graph every point on y f(x) has moved 3
units to the right.
-5
3
?
?
?
3
-5
?
?
?
Notef (x 5) moves the graph -5.
Notef (x - 3) moves the graph 3.
-5
3
?
?
?
x
3
-5
0
y f (x - 3)
y f (x 5)
y f (x)
14
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
y
In this graph every point on y f(x) has moved a
units to the left.
In this graph every point on y f(x) has moved b
units to the right.
Notef (x a) moves the graph -a.
Notef (x - b) moves the graph b.
x
0
-a
b
y f (x - b)
y f (x a)
y f (x)
15
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
1 Write down an equation for each graph.
Click the mouse to see the answers.
y
b
a
x
0
-5
3
y (x 5)2
y (x - 3)2
y x2
16
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
2 Write down an equation for each graph.
Click the mouse to see the answers.
y
a
b
x
0
-3
-5
4
y f (x)
y f (x - 4)
y f (x 3)
17
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
3 Write down an equation for each graph.
Click the mouse to see the answers.
y
a
b
x
-3
8
-5
3
2
0
-11
y 9 - x2
y 9 - (x - 5)2
y 9 - (x 8)2
18
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Section 3
y -f (x ) Reflect in x-axis
19
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
y
?
y f (x)
?
?
x
In this graph every point on y f(x) has been
reflected in the x - axis.
?
y - f (x)
?
20
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
1 Write down an equation for each graph.
Click the mouse to see the answers.
y
y x2
b
y (x - 5)2
x
5
a
y -x2
21
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
2 Write down an equation for each graph.
Click the mouse to see the answers.
y
y f (x)
x
-6
-2
b
y -f (x)
a
y f (x 4)
22
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
3 Write down an equation for each graph.
Click the mouse to see the answers.
y
1
0.5
0
x
90?
180?
360?
270?
-0.5
-1
y sin x?
a
b
y -sin x?
y sin(x - 30)?
23
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Section 4
y f (-x ) Reflect in y-axis
24
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
y
?
?
?
?
x
In this graph every point on y f(x) has been
reflected in the y - axis.
?
y f (x)
y f (-x)
25
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
1 Write down an equation for each graph.
Click the mouse to see the answers.
y
y f (x)
b
y f (-x)
x
a
y -f(x)
26
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
2 Write down an equation for each graph.
Click the mouse to see the answers.
y
5
y f (x)
a
y f (-x)
x
0
b
y f (x) - 5
27
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
3 Write down an equation for each graph.
Click the mouse to see the answers.
y
a
y f (x)
y f (-x)
x
0
y - f (x)
b
28
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Section 5
y kf (x) Stretch/compress vertically
29
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
In this graph everyy - value of y sinx? has
been doubled.
y 2 sin x?
y
?
2
1.5
y sin x?
?
1
?
0.5
?
0
x
90?
270?
360?
180?
?
-0.5
In this graph everyy - value of y sinx? has
been halved.
?
y ½ sin x?
-1
-1.5
?
-2
30
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
y k f (x) , k ? 1
y
y f (x)
0
x
y k f (x) , k ? 1
31
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
1 Write down an equation for each graph.
Click the mouse to see the answers.
y
4
a
y 4 sin x?
3
2
y 2 sin x?
b
1
0
x
-1
y sin x?
-2
-3
-4
32
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
2 Write down an equation for each graph.
Click the mouse to see the answers.
y
3
a
y 4 cos x?
2
y cos x?
1
0
x
-1
y ½ cos x?
b
-2
-3
33
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
3 Write down an equation for each graph.
Click the mouse to see the answers.
y
a
y 1.5 sin x?
1
0
x
b
y sin x?
y 0.5 sin x?
-1
34
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
Section 6
y f (kx) Stretch/compress horizontally
35
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
In this graph everyy - value sin (double x?)
y sin 2x?
y
?
?
1
y sin x?
?
?
0.5
?
?
0
x
270?
360?
90?
180?
? y sin 2x runsthrough 2 cycles in the time y
sinxruns through 1 cycle
-0.5
-1
36
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
In this graph everyy - value sin (half x?)
y sin ½x?
y
?
?
1
?
?
0.5
y sin x?
?
?
0
x
270?
360?
90?
180?
? y sin ½x runsthrough ½ of a cycle in the
time y sinxruns through 1 cycle
-0.5
-1
37
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
y f (2x)
y f (x)
y f (½x)
y
4 cycles
2 cycles
1 cycle
1
How manycycles?
How manycycles?
0
x
How manycycles?
-1
38
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
1 Write down an equation for this graph.
Click the mouse to see the answer.
y
1
y sin x?
x
0
90?
180?
360?
270?
y sin 3x?
-1
39
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
2 Write down an equation for this graph.
Click the mouse to see the answer.
y
1
y sin x?
x
0
540?
720?
180?
900?
1080?
360?
y sin 1/3x?
-1
40
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
3 Write down an equation for this graph.
Click the mouse to see the answer.
y
y sin x?
1
x
0
540?
720?
180?
360?
-1
y sin ½x?
41
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
4 Write down an equation for this graph.
Click the mouse to see the answer.
y
y sin x?
1
0
x
90?
270?
180?
360?
-1
y sin 2x?
42
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
5 Write down an equation for this graph.
Click the mouse to see the answer.
y
y cos x?
1
0
x
90?
270?
180?
360?
-1
y cos 2x?
43
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
6 Write down an equation for this graph.
Click the mouse to see the answer.
y
y cos x?
1
0
x
90?
540?
720?
360?
450?
270?
180?
630?
-1
y cos ½x?
44
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(a) Sketch and identify related
functions and graphs
7 Write down an equation for each graph.
Click the mouse to see the answers.
y
a
y 3 sin x?
3
b
y - sin x?
2
1
0
x
90?
540?
720?
360?
450?
270?
180?
630?
-1
y sin x?
-2
-3
y sin x? - 3
c
45
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
Exponential
and
logarithmic
graphs
46
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
Revision of powers
22 4 32 9 42 16 52 25 102
100 n2 n2
21 2 31 3 41 4 51 5
101 10 n1 n
20 1 30 1 40 1 50 1
100 1 n0 1
2-1
3-1
4-1
5-1
10-1
n-1
2-2
3-2
4-2
10-2
5-2
n-2
47
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
y 2x
x -2 -1 0 1 2 3 4
y 1 2 4 8
16
y log2x
x 1 2 4 8
16
y -2 -1 0 1 2 3 4
48
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
This is called the exponentialfunction with base
2
y
y 2x
Now we will draw the graphs of the 2 functionson
the previous page.
?
16
14
12
10
y x
?
Each of these functionsis the inverse of the
other.
8
y log2x
6
?
?
4
?
?
?
2
?
?
?
?
i.e. you could get eachby reflecting the
otherin the line y x.
?
0
-4
-2
8
10
12
14
16
2
4
6
?
x
?
-2
-4
This is called the logarithmic functionwith
base 2
49
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
1 Write down an equation for each graph.
y
y 3x
16
14
12
10
?
8
Click the mouse to see the answers.
6
y log3x
4
?
?
2
?
?
?
?
?
0
-4
-2
8
10
12
14
16
2
4
6
?
x
?
-2
-4
50
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
2 Write down an equation for each graph.
y
y 5x
32
28
?
Read the scale!
24
20
16
Click the mouse to see the answers.
12
y log5x
8
?
4
?
?
?
?
?
?
0
-8
-4
16
20
24
28
32
4
8
12
?
x
?
-4
-8
51
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(b) Identify exponential and
logarithmic graphs
3 Write down an equation for each graph.
y 10x
y
?
10
Read the scale!
8
6
Click the mouse to see the answers.
4
y log10x
2
?
?
?
?
?
6
0
10
4
8
2
-2
x
?
?
-2
52
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
Composite functions
g (x)
f (x)
g (x)
x
f (g (x))
53
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
f(x) x3 and g(x) 2x -3.Find expressions for
a f(g(x)) b
g(f(x))
Example 1
a
g(x) 2x -3
f(x) x3
g(x)
f(g(x))f(2x - 3)
x
2x - 3
(2x - 3)3
f(g(x)) f(2x -3) (2x - 3)3
54
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
f(x) x3 and g(x) 2x -3.Find expressions for
a f(g(x)) b
g(f(x))
Example 1
b
g(x) 2x -3
f(x) x3
f(x)
g(f(x))g(x3)
x
x3
2x3 - 3
g(f(x)) g(x3) 2x3 - 3
55
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
h(x) 5x and k(x) sin x?.Find expressions for
a h(k(x)) b
k(h(x))
Example 2
a
h(x) 5x
k(x) sin x
k(x)
h(k(x))h(sin x)
x
sin x
5 sin x
h(k(x)) h(sin x) 5 sin x
56
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
h(x) 5x and k(x) sin x?.Find expressions for
a h(k(x)) b
k(h(x))
Example 2
b
k(x) sin x
h(x) 5x
h(x)
k(h(x))k(5x)
x
5x
sin 5x
k(h(x)) k(5x) sin 5x
57
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
g(x) and h(x) x2 -1.Find expressions
for a g(h(x))
b h(g(x))
Example 3
a
g(x)
h(x) x2 -1
g(h(x))g(x2 -1)
h(x)
x
x2 -1
g(h(x)) g(x2 -1)
58
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
g(x) and h(x) x2 -1.Find expressions
for a g(h(x))
b h(g(x))
Example 3
b
h(x) x2 -1
g(x)
h(g(x))
g(x)
x
h(g(x))
59
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
1 Functions f and g are defined byf(x) x2 and
g(x) 3x.Find expressions for a f(g(x))
b g(f(x))
Here are the answers
a f(g(x)) f(3x) (3x)2
9x2
b g(f(x)) g(x2) 3x2
60
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
2 Functions h and k are defined byh(x) 3x and
k(x) cos x?.Find expressions for a h(k(x))
b k(h(x))
Here are the answers
a h(k(x)) h(cos x?) 3 cos x?
b k(h(x)) k(3x) cos 3x?
61
Mathematics 1(Higher)
Outcome 2 Associate functions and graphs
PC(c) Find composite functions of the
form f (g(x)), given f (x) and g(x)
3 Functions g and h are defined byg(x) 2x - 1
and h(x) x3.Find expressions for a g(h(x))
b h(g(x))
Here are the answers
a g(h(x)) g(x3) 2 x3 - 1
b h(g(x)) h(2x - 1) (2x - 1)3