Transcript 6.9 Modeling with polynomial functions

6.9 Modeling with Polynomial
Functions
p. 380
Ex: Write the cubic function whose
graph goes through the points (-2,0),
(0,2), (1,0), and (3,0).
• The 3 x-intercepts (-2,0), (1,0), and (3,0)
will give you the 3 zeros of the cubic.
They will also tell you 3 factors to use
f(x)=a(x+2)(x-1)(x-3).
• Then use the 4th point as x & f(x) values.
2=a(0+2)(0-1)(0-3)
• Now solve for a! 2=6a so, a=1/3
• Answer: f(x)=1/3(x+2)(x-1)(x-3)
Ex: An eqn. for a polynomial function is
f(n)=2n3+n2+2n+1. Show that this function has constant
3rd order differences. (you check the 3rd order diffs.
because it’s a degree 3 polynomial)
• First, write out the first several values; or find f(1), f(2),
f(3),…, f(6).
• f(1)=6 f(2)=25 f(3)=70 f(4)=153 f(5)=286 f(6)=481
Now subtract #s! (left from right)
1st diffs. 19
45
83
133
195
Now subtract #s! (left from right)
2nd diffs
26
38
50
62
Now subtract #s! (left from right)
3rd diffs.
12
12
12
** This is called using finite differences.
Ex:The values of a polynomial function for six
consecutive whole numbers are given below. Write a
polynomial function for f(n).
f(1)= -2, f(2)=2, f(3)=12, f(4)=28, f(5)=50, andf(6)=78
• First, find finite differences. (Stop when the same
number repeats all the way across!)
4
10
16
22
28
6
6
6
6
The 2nd differences are now a constant # across.
This means the function will be a quadratic. (degree 2)
So, use f(n)=an2+bn+c.
Since you must find a, b, & c, you will need to make 3
eqns. with these 3 variables using the first 3 known
values of the function.
Use an2+bn+c=f(n) & f(1)= -2, f(2)=2,
f(3)=12 to write 3 equations.
a(1)2+b(1)+c= -2
a+b+c= -2
a(2)2+b(2)+c=2
4a+2b+c=2
a(3)2+b(3)+c=12
9a+3b+c=12
1  familiar?
1**1 Look
 2 It should! **a 




1


A  inverse
4 2 1 matrices
A for
* B a,
 b,
b &c! *
B   2 to
* Use
solve

9 3 1
c 
 12 
a   3 
b    5
   
c   0 
This means the quadratic is
f(n)=3n2-5n+0 or
f(n)=3n2-5n
Assignment