1. 6.9 Modeling with Polynomial Functions

2. 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)

3. Ex: An eqn. for a polynomial function is f(n)=2n3+n2+2n+1
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
2nd diffs
3rd diffs
** This is called using finite differences.

4. 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!)
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.

5. 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
** Look familiar? It should! **
* Use inverse matrices to solve for a, b, &c! *
This means the quadratic is f(n)=3n2-5n+0 or
f(n)=3n2-5n

6. Assignment