6.9 Modeling with Polynomial Functions

p. 380

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: Write the cubic function whose graph goes through the points (-2,0), (0,2), (1,0), and (3,0).

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)=481Now 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.

First, find finite differences. (Stop when the same number repeats all the way across!)

4 10 16 22 28

6 6 6 6The 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.

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

a(1)2+b(1)+c= -2 a+b+c= -2a(2)2+b(2)+c=2 4a+2b+c=2a(3)2+b(3)+c=12 9a+3b+c=12

** Look familiar? It should! *** Use inverse matrices to solve for a, b, &c! *

Use an2+bn+c=f(n) & f(1)= -2, f(2)=2, f(3)=12 to write 3 equations.

1 3 91 2 41 1 1

A

12 2 2

B

cba

BA *1

0 5

3

cba This means the quadratic is

f(n)=3n2-5n+0 or

f(n)=3n2-5n

f(1)= -2, f(2)=2, f(3)=12, f(4)=28, f(5)=50, and f(6)=78 1. [STAT] [1] this is the edit key 2. enter in all the x values in L1 3.enter in all the y values in L2 4.[STAT] arrow to CALC 5. Find the regression you need. ***Same as the finite order

difference you found.◦ linReg (ax +b) is linear degree 1◦ QuadReg is quadradic degree 2◦ CubicReg is cubic degree 3◦ QuartReg is quartic degree 4.6. Write the equation in standard form

ORUse an2+bn+c=f(n) and all points in

the calculator

When given special points ◦– the x intercepts- are the special points Writing the equation in factored form is

the fastest. F(x) = a (x-p)(x – q) … Substitute in x and y the xintercepts

known as the zero’s and solve for a. Last step leave in F(x) = a (x-p)(x – q)

…

Review

Given many points not the x intercepts.

The answer may be in factored form in the book.

No Worries. They are equivalent!!!!

1.Find the finite difference.

2. Then use the calculator.

STAT CALC It’s the fastest. And it is in

Standard form.

Review

AssignmentDon’t Forget you can use rref in the math for matrices as well.