1. Polynomial Functions and Models
Lesson 4.2

2. Review General polynomial formula
a0, a1, … ,an are constant coefficients
n is the degree of the polynomial
Standard form is for descending powers of x
anxn is said to be the “leading term”

3. Turning Points and Local Extrema
A point (x, y) on the graph
Located where graph changes
from increasing
to decreasing
(or vice versa) •

4. Family of Polynomials Constant polynomial functions
f(x) = a
Linear polynomial functions
f(x) = m x + b
Quadratic polynomial functions
f(x) = a x2 + b x + c

5. Family of Polynomials Cubic polynomial functions
f(x) = a x3 + b x2 + c x + d
Degree 3 polynomial
Quartic polynomial functions
f(x) = a x4 + b x3 + c x2+ d x + e
Degree 4 polynomial

6. Compare Long Run Behavior
Consider the following graphs:
f(x) = x4 - 4x3 + 16x - 16
g(x) = x4 - 4x3 - 4x2 +16x
h(x) = x4 + x3 - 8x2 - 12x
Graph these on the window -8 < x < 8       and      0 < y < 4000
Decide how these functions are alike or different, based on the view of this graph

7. Compare Long Run Behavior
From this view, they appear very similar

8. Contrast Short Run Behavior
Now Change the window to be -5 < x < 5   and   -35 < y < 15
How do the functions appear to be different from this view?

9. Contrast Short Run Behavior
Differences?
Real zeros
Local extrema
Complex zeros
Note: The standard form of the polynomials does not give any clues as to this short run behavior of the polynomials:

10. Factored Form Consider the following polynomial:
p(x) = (x - 2)(2x + 3)(x + 5)
What will the zeros be for this polynomial?
x = 2
x = -3/2
x = -5
How do you know?
We see the product of two values
a * b = 0 We know that either a = 0 or b = 0 (or both)

11. Factored Form
Try factoring the original functions f(x), g(x), and h(x) 
(enter    factor(y1(x)) 
what results do you get?

12. Local Max and Min
For now the only tools we have to find these values is by using the technology of our calculators:

13. Multiple Zeros Given We say the degree = n
With degree = n, the function can have up to n different real zeros
Sometimes the zeros are repeated, as seen in y1(x) and y3(x) below

14. Multiple Zeros
Look at your graphs of these functions, what happens at these zeros?
Odd power, odd number of duplicate roots => inflection point at root
Even power, even number of duplicate roots => tangent point at root

15. Linear Regression
Used in section previous lessons to find equation for a line of best fit
Other types of regression are available

16. Polynomial Regression
Consider the lobster catch (in millions of lbs.) in the last 30 some years
Enter into Data Matrix
Year
1970
1975
1980
1985
1990
1995
2000 t 5 10 15 20 25 30 35
Lobster 17 19 22 27 36 56

17. Viewing the Data Points
Specify the plot F2,
X's from C1, Y's from C2
View the graph
Check Y= screen, use Zoom-Data

18. Polynomial Regression
Try for 4th degree polynomial

19. Other Technology Tools
Excel will also do regression
Plot data as (x,y) ordered pairs
Right click on data series
Choose Add Trend Line

20. Other Technology Tools
Use dialog box to specify regression
Try Others

21. Assignment Lesson 4.2A Page 247 Exercises 1 – 41 odd Lesson 4.2B