Expert: Paul Klarreich - 9/28/2008

Question
hi I'm studying grade 12 advanced functions
The question:
sketch the graph of the function. Determine the equation of the function.
Describe the function including restricted and non restricted domainsnd
range, end behaviour, max or min values, intecepts

Calculate the rate of change and determine how the values are changing.
table of values given:
time(mins)    population
0,                   800
1,                   799
2,                   782
3,                   737
4,                   652
5,                   515
6,                   314

I drew the graph but I'm having trouble determine the equation


Answer
Questioner:   Vanessa
Category:  Advanced Math
Private:  No
 
Subject:  polynomial functions
Question:  hi I'm studying grade 12 advanced functions
The question:
sketch the graph of the function. Determine the equation of the function. Describe the function including restricted and non restricted domain and range, end behaviour, max or min values, intecepts

Calculate the rate of change and determine how the values are changing.
table of values given:
time(mins)    population
0,                   800
1,                   799
2,                   782
3,                   737
4,                   652
5,                   515
6,                   314

I drew the graph but I'm having trouble determine the equation
.......................................................
Hi, Vanessa,

This is a cutie.  (I'm sure you are, too.)   

Unfortunately, there are an infinite number of possible polynomial that satisfy the data, so we have to make a assumption.  What assumption?  Look at these computed differences.

WARNING: THIS DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL DIFFICULT TO VIEW ON CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.


X      f(x)   D(f(x))  DD(f)  DDD(f)
1   800         
2   799   -1      
3   782   -17   -16   
4   737   -45   -28   -12
5   652   -85   -40   -12
6   515   -137   -52   -12
7   314   -201   -64   -12

Do you see how each was computed?  Now the fact that the 'third differences' are constant suggests that this is a third-degree equation.

It has the form:

P(x) =  ax^3 + bx^2 + cx + d
and we can do some other fancy footwork.  If we assume the DDD column must always be -12, we can fill in:

X      f(x)   D(f(x))  DD(f)  DDD(f)
0       797 <<
1   800    3 <<      
2   799   -1    -4 <<   
3   782   -17   -16   -12 <<
4   737   -45   -28   -12
5   652   -85   -40   -12
6   515   -137   -52   -12
7   314   -201   -64   -12




(If you were doing calculus, I could do some other stuff, but you said you were not.  Good for you.)

Now all you have to do (all, he says....) is plug in a few values: (four of them, to be specific, because we need four values -- a,b,c,d.

P() =  a()^3 + b()^2 + c() + d

P(0) =  a(0)^3 + b(0)^2 + c(0) + d = 797
P(1) =  a(1)^3 + b(1)^2 + c(1) + d = 800
P(2) =  a(2)^3 + b(2)^2 + c(2) + d = 799
P(3) =  a(3)^3 + b(3)^2 + c(3) + d = 782

So we get  d = 797 right away:

P(1) =  a(1)^3 + b(1)^2 + c(1) + 797 = 800
P(2) =  a(2)^3 + b(2)^2 + c(2) + 797 = 799
P(3) =  a(3)^3 + b(3)^2 + c(3) + 797 = 782

  a  + b  + c = 3    (A)
 8a + 4b + 2c = 2    (B)
 4a + 2b +  c = 1    (B')
27a + 9b + 3c = -15   (C)
 9a + 3b + c = -5  (C')

Should not be too hard to solve:

  a  + b  + c = 3    (A)
 4a + 2b  + c = 1    (B')
 9a + 3b  + c = -5  (C')

combine A,B:
- a  - b  - c = -3    (-A)
 4a + 2b  + c = 1    (B')
-----------------------------
       3a + b = -2   (D)

combine A,C:
- a  - b  - c = -3    (-A)
 9a + 3b  + c = -5  (C')
------------------------------
 8a  + 2b  = -8   (E)
 4a  + b = -4

  3a + b = -2  (D)
  4a + b = -4 (E)
----------------------- subtract
******   a = -2 *******
Back-sub in E:

-8 + b = -4
*****    b = 4  ******


Back-sub in A:

 a  + b  + c = 3    (A)
- 2 +  4 + c = 3
      2 + c = 3
********  c = 1 *********

OKAY! our polynomial is:

P(x) = -2x^3 + 4x^2 + x + 797

I think I'll leave the rest to you.  If you need more help, let me know.