By: Emma Stevens

Teaching Transformations of Functions using Music

Objectives

Students will be able to analyze nonlinear relationships to explain how the change in one variable results in the change of another.

They will explain how that change affects the graphs and then apply the changes to a melody of notes.

Previous knowledge

Students should have an understanding about a function f(x).

They should understand that a function inputs x values (the domain or the pre-image) and then outputs f(x) values or y values (the range or image).

Keys of the Piano

Original Tune Tablex f(x)

0 0

1 0

2 4

3 4

4 5

5 5

6 4

7 3

8 3

9 2

10 2

11 1

12 1

13 0

x f(x)

14 4

15 4

16 3

17 3

18 2

19 2

20 1

21 4

22 4

23 3

24 3

25 2

26 2

27 1

x f(x)

28 0

29 0

30 4

31 4

32 5

33 5

34 4

35 3

36 3

37 2

38 2

39 1

40 1

41 0

Original Tune Graph

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

Things to Keep in MindThe function repeats its self; so that f(42)= f(0)

=0, f(43)= f(1)= 0, f(44)=f(2)=4 and so on.The x-values are mod 42

Also this does not take in consideration of flats and sharps in new keys so the song with the transformation may not sound right.i.e. the notes are in (mod 7) – The y-valuesSo 0 = 7 = C, 1 = 8 = D, 2 = 9 = E, 3 = 10 = F….

The beat is the same throughout the song: i.e. each note is the same count (it is a note a beat)-meaning it doesn’t take in account for rhythm

Slowing Down a Tune

(i.e. stretching a function)

g(x) = f(1/2x) This transformation inputs x values, then multiply each x

value by 1/2 and the output is f(1/2x).when x=0, g(0) = f(1/2*0) = f(0) =0when x=2, g(2) = f(1/2*2) = f(1) =0when x=4, g(4) = f(1/2*4) = f(2) = 4And so on…

But some values of x will not have a corresponding y value(because the x values and f(x) values are both integers):when x=1, g(1) = f(1/2*1)= f(1/2)when x=3, g(3) = f(1/2*3) = f(3/2)And so on…

xB

f1

Stretched Tune Tablex f(x

)1/2x g(x)

0 0 0 0

1 0 0.5

2 4 1 0

3 4 1.5

4 5 2 4

5 5 2.5

6 4 3 4

7 3 3.5

8 3 4 5

9 2 4.5

10 2 5 5

11 1 5.5

12 1 6 4

13 0 6.5

x f(x) 1/2x g(x)

14 4 7 3

15 4 7.5

16 3 8 3

17 3 8.5

18 2 9 2

19 2 9.5

20 1 10 2

21 4 10.5

22 4 11 1

23 3 11.5

24 3 12 1

25 2 12.5

26 2 13 0

27 1 13.5

x f(x) 1/2x g(x)

28 0 14 4

29 0 14.5

30 4 15 4

31 4 15.5

32 5 16 3

33 5 16.5

34 4 17 3

35 3 17.5

36 3 18 2

37 2 18.5

38 2 19 2

39 1 19.5

40 1 20 1

41 0 20.5

Stretched Tune Graph

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

f(x) g(x)

Speeding Up a Tune(i.e. condensing a function ) We will call this transformation h(x) = f(2x) This transformation inputs the x values, multiplies

each x value by 2 and outputs f(2x) values.when x=0, h(0)=f(2*0)= f(0)= 0when x=1, h(1)=f(2*1)= f(2)= 4when x=2, h(2)=f(2*2)= f(4)= 5And so on…

Notice that f(1), f(3), f(5) and so on don’t exist in this transformation; therefore only half of the notes are in the function

)(Bxf

Condensed Tune Tablex f(x

) 2x h(x)

0 0 0 0

1 0 2 4

2 4 4 5

3 4 6 4

4 5 8 3

5 5 10 2

6 4 12 1

7 3 14 4

8 3 16 3

9 2 18 2

10 2 20 1

11 1 22 4

12 1 24 3

13 0 26 2

x f(x) 2x h(x)

14 4 28 0

15 4 30 4

16 3 32 5

17 3 34 4

18 2 36 3

19 2 38 2

20 1 40 1

21 4 42=0 0

22 4 44=2 4

23 3 46=4 5

24 3 48=6 4

25 2 50=8 3

26 2 52=10 2

27 1 54=12 1

x f(x) 2x h(x)

28 0 56=14 4

29 0 58=16 3

30 4 60=18 2

31 4 62=20 1

32 5 64=22 4

33 5 66=24 3

34 4 68=26 2

35 3 70=28 0

36 3 72=30 4

37 2 74=32 5

38 2 76=34 4

39 1 78=36 3

40 1 80=38 2

41 0 82=40 1

Condensed Tune Graph

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

f(x) h(x)

Shift the Tune to Play a Round(i.e. horizontal shift of a function)

j(x) = f(x+4) This transformation inputs the x values, adds 4 to

each x value and outputs f(x+4) values.when x=1, j(1)= f(1+4)= f(5)= 5when x=2, j(2)=f(2+4)= f(6)= 5when x=3, j(3)=f(3+4)= f(7)= 4And so on…

Notice that the starting note of the function is actually the fifth note of the original function and f(1), f(2), f(3), and f(4) have been taken off the front of the song and attached to back of the song.

)( Cxf

Horizontal Shift Tune Tablex f(x

) x+4 j(x)

0 0 4 5

1 0 5 5

2 4 6 4

3 4 7 3

4 5 8 3

5 5 9 2

6 4 10 2

7 3 11 1

8 3 12 1

9 2 13 0

10 2 14 4

11 1 15 4

12 1 16 3

13 0 17 3

x f(x) x+4 j(x)

14 4 18 2

15 4 19 2

16 3 20 1

17 3 21 4

18 2 22 4

19 2 23 3

20 1 24 3

21 4 25 2

22 4 26 2

23 3 27 1

24 3 28 0

25 2 29 0

26 2 30 4

27 1 31 4

x f(x) x+4 j(x)

28 0 32 5

29 0 33 5

30 4 34 4

31 4 35 3

32 5 36 3

33 5 37 2

34 4 38 2

35 3 39 1

36 3 40 1

37 2 41 0

38 2 42=0 0

39 1 43=1 0

40 1 44=2 4

41 0 45=3 4

Horizontal Shift Tune Graph

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

f(x) j(x)

Change Keys of a Tune(i.e. vertical shift of a function)

k(x) = f(x) + 2 This transformation inputs the x values, adds 2

to each f(x) value and outputs f(x)+2 values.when x=1, k(1)=f(1)+2= 0+2=2when x=2, k(2)=f(2)+2= 0+2=2when x=3, k(3)=f(3)+2=4+2 =6And so on…

Notice that the starting note of the function is two notes higher than the original function

Dxf )(

Vertical Shift Tune Tablex f(x

) k(x)

0 0 2

1 0 2

2 4 6

3 4 6

4 5 7

5 5 7

6 4 6

7 3 5

8 3 5

9 2 4

10 2 4

11 1 3

12 1 3

13 0 2

x f(x) k(x)

14 4 6

15 4 6

16 3 5

17 3 5

18 2 4

19 2 4

20 1 3

21 4 6

22 4 6

23 3 5

24 3 5

25 2 4

26 2 4

27 1 3

x f(x) k(x)

28 0 2

29 0 2

30 4 6

31 4 6

32 5 7

33 5 7

34 4 6

35 3 5

36 3 5

37 2 4

38 2 4

39 1 3

40 1 3

41 0 2

Vertical Shift Tune Graph

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

7

8

f(x) k(x)

Stretching the Notes Apart(i.e. vertical stretch) r(x) = 3f(x)The transformation that inputs x values and

outputs 3f(x) values; Therefore:When x=1, r(1)=3f(1)= 3(0)=0When x=2, r(2)=3f(2)= 3(0)=0When x=3, r(3)=3f(3)= 3(4)=12And so on…

This song stretches the original notes further apart, so that they have a greater distance between them.

)(xAf

x f(x) r(x)

0 0 0

1 0 0

2 4 12

3 4 12

4 5 15

5 5 15

6 4 12

7 3 9

8 3 9

9 2 6

10 2 6

11 1 3

12 1 3

13 0 0

x f(x) r(x)

14 4 12

15 4 12

16 3 9

17 3 9

18 2 6

19 2 6

20 1 3

21 4 12

22 4 12

23 3 9

24 3 9

25 2 6

26 2 6

27 1 3

x f(x) 4(x)

28 0 0

29 0 0

30 4 12

31 4 12

32 5 15

33 5 15

34 4 12

35 3 9

36 3 9

37 2 6

38 2 6

39 1 3

40 1 3

41 0 0

Vertical Stretch Tune Table

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 410

2

4

6

8

10

12

14

16

f(x) r(x)

Vertical Stretch Tune Graph

Squishing the Notes Together(i.e. vertical condensing)

)(1

xfA

s(x) =(1/2) f(x) This transformation inputs x values, then multiply each f(x)

value by 1/2 and the output is (1/2)f(x).when x=0, s(0) = (1/2)f(0) = (1/2)(0) = 0when x=2, s(1) = (1/2)f(2) = (1/2)0 = 0when x=4, s(2) = (1/2)f(2) = (1/2)4 = 2And so on…

But some values of x will not have a corresponding y value(because the x values and f(x) values are both integers):when x=4, s(4) = (1/2)f(4)= (1/2)5 = 2.5when x=11, s(11) = (1/2)f(11) = (1/2)1=0.5And so on…

x f(x) s(x)

0 0 0

1 0 0

2 4 2

3 4 2

4 5

5 5

6 4 2

7 3

8 3

9 2 1

10 2 1

11 1

12 1

13 0 0

x f(x) s(x)

14 4 2

15 4 2

16 3

17 3

18 2 1

19 2 1

20 1

21 4 2

22 4 2

23 3

24 3

25 2 1

26 2 1

27 1

x f(x) s(x)

28 0 0

29 0 0

30 4 2

31 4 2

32 5

33 5

34 4 2

35 3

36 3

37 2 1

38 2 1

39 1

40 1

41 0 0

Vertical Condensing Tune Table

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 410

1

2

3

4

5

6

f(x) s(x)

Vertical Condensing Tune Graph

Playing a Tune Backwards(i.e. a flip over the y-axis)This transformation inputs x values and outputs f(-x) values

Therefore:

When x= 1, m(1)=f(-1)=f(41) = 0

When x= 2, m(2)=f(-2))=f(40)= 1

When x= 3, m(3)=f(-3)= f(39)= 1

And so on…

m(x) = f(-x)

The we will actually play the last note first

i.e. the song is backwards)( xf

Flip over y-axis Tune Table x f(x

) -xm(x

)0 0 0=42 0

1 0 -1=41

0

2 4-

2=401

3 4-

3=391

4 5-

4=382

5 5-

5=372

6 4-

6=363

7 3-

7=353

8 3-

8=344

9 2-

9=335

10 2-

10=32

5

11 1-

11=31

4

12 1-

12=30

4

13 0-

13=29

0

x f(x) -x

m(x)

14 4-

14=28

0

15 4-

15=27

1

16 3-

16=26

2

17 3-

17=25

2

18 2-

18=24

3

19 2-

19=23

3

20 1-

20=22

4

21 4-

21=21

4

22 4-

22=20

1

23 3-

23=19

2

24 3-

24=18

2

25 2-

25=17

3

26 2-

26=16

3

27 1-

27=15

4

x f(x) -x

m(x)

28 0-

28=14

4

29 0-

29=13

0

30 4-

30=12

1

31 4-

31=11

1

32 5-

32=10

2

33 5-

33=9 2

34 4-

34=8 3

35 3-

35=7 3

36 3-

36=6 4

37 2-

37=5 5

38 2-

38=4 5

39 1-

39=3 4

40 1-

40=2 4

41 0-

41=1 0

-50 -40 -30 -20 -10 0 10 20 30 40 500

1

2

3

4

5

6

m(x) f(x)

Literal Flip Over the Y-Axis

Flip over y-axis Tune Graph

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

m(x) f(x)

Turning the Tune Upside-down(i.e. flip over the x-axis) p(x) = - f(x)The transformation that inputs x values and

outputs -f(x) values; Therefore:When x=1, p(1)=-f(1)= 0When x=2, p(2)=-f(2)= 0When x=3, p(3)=-f(3)= - 4And so on…

This song has completely different notes than the original tune and the notes are lower.

)(xf

Flip over x-axis Tune Tablex f(x

) p(x)

0 0 0

1 0 0

2 4 -4

3 4 -4

4 5 -5

5 5 -5

6 4 -4

7 3 -3

8 3 -3

9 2 -2

10 2 -2

11 1 -1

12 1 -1

13 0 0

x f(x) p(x)

14 4 -4

15 4 -4

16 3 -3

17 3 -3

18 2 -2

19 2 -2

20 1 -1

21 4 -4

22 4 -4

23 3 -3

24 3 -3

25 2 -2

26 2 -2

27 1 -1

x f(x) p(x)

28 0 0

29 0 0

30 4 -4

31 4 -4

32 5 -5

33 5 -5

34 4 -4

35 3 -3

36 3 -3

37 2 -2

38 2 -2

39 1 -1

40 1 -1

41 0 0

Flip over x-axis Tune Graph

0 5 10 15 20 25 30 35 40 45

-6

-4

-2

0

2

4

6

f(x) p(x)

Putting it All Together!

DCBxfA )(

Summarization of Transformations

→Shift the function horizontal B units→Shift the function to the vertical C units

→Reflection of the function over the x-axis→Reflection of the function over the y- axis

→ Condenses the function Horizontal

→ Stretches the function Horizontal

xB

f1

Bxf

Cxf

Dxf

xf

xf

xfA

1

xAf → Stretches the function Vertical

→ Condenses the function Vertical

Special Thanks to Betty Clifford for advising me

though this project.