basic skills in higher mathematics

Basic Skills in

Higher Mathematics

Robert GlenAdviser in Mathematics

Mathematics 1(H)Outcome 2

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

Functions and

graphs

y

x



y

x

PC Index

PC(a) Related functions and graphs

PC(b) Exponential and logarithmic graphs

PC(c) Composite functions

Click on the one you want



PC(a) Sketch and identify related functions and graphs

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

5 y = kf (x) Stretch/compress vertically 6 y = f (kx) Stretch/compress horizontally

4 y = f (-x) Reflect in y-axis




Section 1

y = f (x) a Slide up/down




x

y

0

-2

+5

y = f (x) + 5

y = f (x)

y = f (x) - 2

In this graph every point on y = f(x) has moved up 5 units.

In this graph every point on y = f(x) has moved down 2 units.

5

+5

+5

-2

-2

-2




x

y y = f (x) + a

y = f (x)

y = f (x) - b

In this graph every point on y = f(x) has moved up a units.

In this graph every point on y = f(x) has moved down b units.

-b

+a




x

y

a

y = x2

b3

-1

Here are the answers

a

b

y = x2 + 3

y = x2 - 1

0

1 Write down an equation for each graph.




x

y

a

y = x3 - 5x

b


a

b

y = x3 - 5x + 9

y = x3 - 5x - 8

9

-8

0





x

y

a

y = f (x)

b


a

b

y = f (x) + 4

y = f (x) - 6

8

-6

0

6

4

2

-4

-2





Continue with Section 2

End of Section 1




Section 2

y = f (x a) Slide left/right




x

y

3-5 0

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.

y = f (x + 5)y = f (x)

y = f (x - 3)

+3

+3

+3-5

-5

-5

Notef (x - 3) moves the graph +3.

Notef (x + 5) moves the graph -5.




x

y

0

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.

y = f (x + a)y = f (x)

y = f (x - b)

Notef (x - b) moves the graph +b.

Notef (x + a) moves the graph -a.

+b-a





x

y

y = (x + 5)2

y = x2y = (x - 3)2

a b

3-5 0

Click the mouse to see the answers.





x

y

a b

4-3-5

y = f (x + 3)y = f (x)

y = f (x - 4)

0






x

y

a b

8-3-5

y = 9 - (x + 8)2 y = 9 - x2 y = 9 - (x - 5)2

0 32-11






End of Section 2




Section 3

y = -f (x ) Reflect in x-axis




y = f (x)

y = - f (x)

x

y

In this graph every point on y = f(x) has been reflected in the x - axis.





x

y

a

by = x2

y = -x2

y = (x - 5)2

5






x

y

ab

y = f (x)

-6 -2


y = -f (x)

y = f (x + 6)





x

y

a by = sin x

y = -sin x y = sin(x - 30)


0.5

1

0

-0.5

-1

90 180 270 360





End of Section 3




Section 4

y = f (-x ) Reflect in y-axis




y = f (x)y = f (-x)

x

y

In this graph every point on y = f(x) has been reflected in the y - axis.





x

y

a

by = f (x)

y = -f(x)

y = f (-x)






x

y

a

b

y = f (x)


y = f (-x)

y = f (x) - 5

5

0





x

y

a

b

y = f (x)


y = f (-x)

y = - f (x) 0





End of Section 4




Section 5

y = kf (x) Stretch/compress

vertically




x

y

0.5

1

0

-0.5

-1

-1.5

-2

2

1.5 y = sin x

y = 2 sin x

y = ½ sin x

In this graph everyy - value of y = sinx has been doubled.

In this graph everyy - value of y = sinx has been halved.

90 270180 360




x

y

y = f (x)

y = k f (x) , k 1

y = k f (x) , k 1

0





x

y

1

2

0

-1

-2

-3

-4

4

3

y = sin x

y = 2 sin x

a

b

y = 4 sin x






x

y

1

2

0

-1

-2

-3

3

y = cos x

y = ½ cos x

a

b

y = 4 cos x






x

y

1

0

-1y = sin xy = 0.5 sin x

a

b

y = 1.5 sin x






End of Section 5




Section 6

y = f (kx) Stretch/compress

horizontally




x

y

y = sin x

y = sin 2xIn this graph everyy - value = sin (double x)

90 270180 360

-1

1

-0.5

0.5

0

y = sin 2x° runsthrough 2 cycles in the time y = sinx°runs through 1 cycle




x

y

y = sin x

y = sin ½xIn this graph everyy - value = sin (half x)

90 270180 360

-1

1

-0.5

0.5

0

y = sin ½x° runsthrough ½ of a cycle in the time y = sinx°runs through 1 cycle




x

y

-1

1

0

y = f (x) y = f (½x)y = f (2x)

How manycycles?

How manycycles?

How manycycles?

2 cycles4 cycles 1 cycle

1 Write down an equation for this graph.




Click the mouse to see the answer.

1

0

-1

x

y

y = sin x

90 270180 360

y = sin 3x






1

0

-1

x

y

y = sin x

540 720180 360

y = sin 1/3x

900 1080






1

0

-1

x

y y = sin x

540 720180 360

y = sin ½x






1

0

-1

x

y y = sin x

90 270180 360

y = sin 2x






1

0

-1

x

y y = cos x

90 270180 360

y = cos 2x






1

0

-1

x

y y = cos x

90 270180 630

y = cos ½x

540 720360 450






1

0

-1x

y

90 270180 630540 720360 450

3

-2

-3

2

y = sin x

b

a

c

y = - sin x

y = 3 sin x

y = sin x - 3




Now do Sections A1 and A2 on pages15 - 17 of the Basic Skills booklet

End of Section 6 and PC(a)



PC(b) Identify exponential and logarithmic graphs




21 = 2 31 = 3 41 = 4 51 = 5 101 = 10 n1 = n

22 = 4 32 = 9 42 = 16 52 = 25 102 = 100 n2 = n2

20 = 1 30 = 1 40 = 1 50 = 1 100 = 1 n0 = 1

21

31

41

51

101

n1

41

91

161

251

1001

21

n2-2 = 3-2 = 4-2 = 5-2 = 10-2 = n-2 =

2-1 = 3-1 = 4-1 = 5-1 = 10-1 = n-1 =

Revision of powers




y = 2x

x -2 -1 0 1 2 3 4

y 1 2 4 8 1641

21

y = log2x

y -2 -1 0 1 2 3 4

x 1 2 4 8 1641

21




y = 2x

08 10 12 14 16

-4

2 4 6-2

-4 -2

16

12

14

8

10

4

6

2

y = log2x

Now we will draw the graphs of the 2 functionson the previous page.

This is called the exponentialfunction with base 2

This is called the logarithmic functionwith base 2

Each of these functionsis the inverse of the other.

i.e. you could get eachby reflecting the otherin the line y = x.

y = x

y

x




y = 3x

08 10 12 14 16

-4

2 4 6-2

-4 -2

16

12

14

8

10

4

6

2

y = log3x

y

x






y = 5x

016 20 24 28 32

-8

4 8 12-4

-8 -4

32

24

28

16

20

8

12

4

y = log5x

y

x



Read the

scale!




y = 10x

1064 82

-2

-2

10

8

4

y = log10x

y

x



2

6

Read the

scale!

0




Now do Sections B1 and B2 on pages23-24 of the Basic Skills booklet

End of PC(b)



PC(c) Find composite functions of the form f (g(x)), given f (x) and g(x)

Composite functions

g (x)

x g (x) f (g (x))

f (x)




f(x) = x3 and g(x) = 2x -3.Find expressions for a f(g(x)) b g(f(x))

a g(x) = 2x -3

g(x)

f(x) = x3

f(g(x))f(2x - 3)x 2x - 3 (2x - 3)3

f(g(x)) = f(2x -3) = (2x - 3)3

Example 1




b g(x) = 2x -3

f(x)

f(x) = x3

g(f(x))g(x3)x x3 2x3 - 3

g(f(x)) = g(x3) = 2x3 - 3

f(x) = x3 and g(x) = 2x -3.Find expressions for a f(g(x)) b g(f(x))

Example 1




h(x) = 5x and k(x) = sin x.Find expressions for a h(k(x)) b k(h(x))

a h(x) = 5x

k(x)

k(x) = sin x°

h(k(x))h(sin x°)x sin x° 5 sin x°

h(k(x)) = h(sin x°) = 5 sin x°

Example 2




b h(x) = 5x

h(x)

k(x) = sin x°

k(h(x))k(5x)x 5x sin 5x°

k(h(x)) = k(5x) = sin 5x°

h(x) = 5x and k(x) = sin x.Find expressions for a h(k(x)) b k(h(x))

Example 2

g(x) =




a

h(x) = x2 -1

h(x)g(h(x))g(x2 -1)x

g(h(x)) = g(x2 -1) =

g(x) = and h(x) = x2 -1.

Find expressions for a g(h(x)) b h(g(x))

Example 3x1

x1

x2 -1 11

2 x

11

2 x

g(x) =




bh(x) = x2 -1

g(x)h(g(x))

x

h(g(x)) = =

g(x) = and h(x) = x2 -1.

Find expressions for a g(h(x)) b h(g(x))

Example 3x1

x1

x1

11

2

x

xh

11

12

x

xh

1




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))


a f(g(x)) = f(3x) = (3x)2

= 9x2

b g(f(x)) = g(x2) = 3x2




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))


a h(k(x)) = h(cos x) = 3 cos x

b k(h(x)) = k(3x) = cos 3x




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))


a g(h(x)) = g(x3) = 2 x3 - 1

b h(g(x)) = h(2x - 1) = (2x - 1)3




Now do Section C1 on page29 of the Basic Skills booklet

End of PC(c)

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