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Basic Skills in Higher Mathematics. Mathematics 1(H) Outcome 2. Robert Glen Adviser in Mathematics. Mathematics 1(Higher). Outcome 2 Associate functions and graphs. y. Functions and graphs. x. Mathematics 1(Higher). Outcome 2 Associate functions and graphs. y. PC Index.

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Basic Skills in Higher Mathematics

Mathematics 1(H)Outcome 2

Robert GlenAdviser in Mathematics

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

y

Functions and graphs

x

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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

Related functions and graphs

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

2y = f (x a) Slide left/right

3y = -f (x) Reflect in x-axis

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

5y = kf (x) Stretch/compress vertically

6 y = f (kx) Stretch/compress horizontally

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

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

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 aunits.

y

y = f (x) + a

y = f (x)

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

+a

y = f (x) - b

x

-b

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

1Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

2Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

3Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

End of Section 1

Continue with Section 2

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

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

-5

3

0

y = f (x - 3)

y = f (x + 5)

y = f (x)

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 aunits to the left.

In this graph every point on y = f(x) has moved bunits 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)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

1Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

2Write 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)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

3Write down an equation for each graph.

Click the mouse to see the answers.

y

a

b

x

-3

8

3

2

-5

-11

0

y =9 - (x - 5)2

y = 9 - x2

y =9 - (x + 8)2

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

End of Section 2

Continue with Section 3

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

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 beenreflected in the x - axis.

y = - f (x)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

1Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

2Write 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 + 6)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

3Write down an equation for each graph.

Click the mouse to see the answers.

y

1

0.5

0

x

90

360

180

270

-0.5

-1

y =sin x

b

a

y =-sin x

y =sin(x - 30)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

End of Section 3

Continue with Section 4

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

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 beenreflected in the y - axis.

y = f (x)

y = f (-x)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

1Write down an equation for each graph.

Click the mouse to see the answers.

y

y = f (x)

y = f (-x)

b

x

a

y =-f(x)

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

2Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

3Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

End of Section 4

Continue with Section 5

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

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 =½sin x

-1

-1.5

-2

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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

1Write down an equation for each graph.

Click the mouse to see the answers.

y

4

a

y =4sin x

3

2

y =2sin x

b

1

0

x

-1

y =sin x

-2

-3

-4

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

2Write down an equation for each graph.

Click the mouse to see the answers.

y

3

a

y =4cos x

2

y =cos x

1

0

x

-1

y =½cos x

b

-2

-3

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

3Write down an equation for each graph.

Click the mouse to see the answers.

y

a

y =1.5sin x

1

0

x

b

y =sin x

y =0.5sin x

-1

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

End of Section 5

Continue with Section 6

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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

In this graph everyy - value = sin (doublex)

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 = sinx°runs through 1 cycle

-0.5

-1

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

In this graph everyy - value = sin (halfx)

y = sin ½x

y

1

0.5

y =sin x

0

x

270

360

90

180

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

-0.5

-1

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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

1Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

2Write down an equation for this graph.

Click the mouse to see the answer.

y

1

y =sin x

x

0

900

180

540

720

1080

360

y =sin 1/3x

-1

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

3Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

4Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

5Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

6Write down an equation for this graph.

Click the mouse to see the answer.

y

y =cos x

1

0

x

90

450

540

720

360

270

180

630

-1

y =cos ½x

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

7Write down an equation for each graph.

Click the mouse to see the answers.

y

a

y =3 sin x

3

y =-sin x

b

2

1

0

x

90

450

540

720

360

270

180

630

-1

y =sin x

-2

-3

y =sin x - 3

c

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(a) Sketch and identify related functions and graphs

End of Section 6 and PC(a)

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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(b) Identify exponential and logarithmic graphs

Exponential

and

logarithmic

graphs

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

10-1=

2-1=

5-1=

4-1=

3-1=

n-1=

4-2=

2-2=

3-2=

10-2=

5-2=

n-2=

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

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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(b) Identify exponential and logarithmic graphs

1Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(b) Identify exponential and logarithmic graphs

2Write 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

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(b) Identify exponential and logarithmic graphs

3Write 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

0

6

-2

2

4

8

10

x

-2

Mathematics 1(Higher)

Outcome 2 Associate functions and graphs

PC(b) Identify exponential and logarithmic graphs

End of PC(b)

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

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))

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 af(g(x))bg(f(x))

Example 1

a

f(x) = x3

g(x) = 2x -3

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

g(x)

x

2x - 3

(2x - 3)3

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

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 af(g(x))bg(f(x))

Example 1

b

g(x) = 2x -3

f(x) = x3

g(f(x))g(x3)

f(x)

x

x3

2x3- 3

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

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) = 5xand k(x) = sin x.Find expressions for ah(k(x))bk(h(x))

Example 2

a

h(x) = 5x

k(x) = sin x°

h(k(x))h(sin x°)

k(x)

x

sin x°

5 sin x°

h(k(x)) = h(sin x°) = 5 sin x°

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) = 5xand k(x) = sin x.Find expressions for ah(k(x))bk(h(x))

Example 2

b

k(x) = sin x°

h(x) = 5x

k(h(x))k(5x)

h(x)

x

5x

sin5x°

k(h(x)) = k(5x) = sin5x°

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 ag(h(x))bh(g(x))

Example 3

a

g(x) =

h(x) = x2 -1

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

h(x)

x2 -1

x

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

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 ag(h(x))bh(g(x))

Example 3

b

h(x) = x2 -1

g(x) =

h(g(x))

g(x)

x

h(g(x)) = =

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)

1Functionsfandg are defined byf(x) = x2and g(x) = 3x.Find expressions foraf(g(x))bg(f(x))

Here are the answers

af(g(x)) = f(3x) = (3x)2= 9x2

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

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)

2Functionsh andk are defined byh(x) = 3x and k(x) = cos x.Find expressions forah(k(x))bk(h(x))

Here are the answers

ah(k(x)) = h(cos x) = 3 cos x

bk(h(x)) = k(3x) = cos 3x

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)

3Functionsg andh are defined byg(x) = 2x - 1and h(x) = x3.Find expressions forag(h(x))bh(g(x))

Here are the answers

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

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

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)

End of PC(c)

Now do Section C1 on page29 of the Basic Skills booklet