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The points that I choose to input into the cubic polynomial function were the

coordinates for 12, 15, 16 and 17 year olds. I chose 12 and 17 year olds because

these were the oldest and youngest of the experiment. raph 1 shows that there is

a possible in!exion between ages 15 and 16.

The points are listed as below"

#12, 2$%& 'aximum (alue

#15, 1)*& +ossible in!exion point

#16, 127& +ossible in!exion point

#17, 12*& 'inimum (alue

The formula for a cubic polynomial function is gi(en as"

y=a x3

+b x2

+cx+d

xwill be dened as the age of the sub-ect ranging from 12 to 17 years old

ywill be dened as the a(erage fundamental freuency for non/spontaneous speech

in 0ert #0&

sing the points that I chose for non/spontaneous speech, I substituted the x and y

(alues into the formula"

122

+c (12 )+d12

3

+b

248=a

152

+c (15 )+d15

3

+b

130=a

162

+c (16 )+d16

3+b

127=a

17

2

+c (17 )+d17

3

+b

120=a

sing the simultaneous euation sol(er in the 34"

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a=113

60

b=522

5

c=98471

60

d=8739

This is the resulting function for non/spontaneous speech

y=113

60x

3+522

5x

298471

60x+8739

The points I chose for the non/spontaneous speech model were"

#12, 21*, 12&

#16, 121&

#17, 11$&

sing the coordinates from abo(e, I substituted the x and y (alues into the formula"

122

+c (12 )+d12

3

+b

210=a

152

+c (15 )+d15

3

+b

129=a

162

+c (16 )+d16

3

+b

121=a

172

+c (17 )+d173+b

114=a

sing the simultaneous euation sol(er in the 34"

a=17

20

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b=413

10

c=13509

20

d=3837

This is the resulting function for spontaneous speech

y=17

20x

3

+413

10x

2

13509

20x+3837