Polynomial Functions and Models Lesson 4.2 Review General polynomial formula P( x) an x an1x n n1 ...
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Transcript Polynomial Functions and Models Lesson 4.2 Review General polynomial formula P( x) an x an1x n n1 ...
Polynomial Functions
and Models
Lesson 4.2
Review
General polynomial formula
P( x) an x an1x
n
n1
... a1x a0
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”
Turning Points and Local
Extrema
Turning point
A point (x, y) on the graph
Located where graph changes
from increasing
to decreasing
(or vice versa)
•
•
Family of Polynomials
Constant polynomial functions
Linear polynomial functions
f(x) = a
f(x) = m x + b
Quadratic polynomial functions
f(x) = a x2 + b x + c
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
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
Compare Long Run Behavior
From this view, they appear very similar
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?
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:
Assignment
Lesson 4.2A
Page 270
Exercises 1 – 38 odd
Piecewise Defined Functions
4 x if x 4
2
f ( x) x 2if 4 x 2
4 x 2 if 2 x
Consider
Sketch graph
Use calculator to display
Consider if f(x) is continuous
Linear Regression
Used in section previous lessons to find
equation for a line of best fit
Other types of
regression are
available
Polynomial Regression
Consider the lobster catch (in millions of lbs.) in
the last 30 some years
Year
1970
1975
1980
1985
1990
1995
2000
t
5
10
15
20
25
30
35
Lobster
17
19
22
20
27
36
56
Enter into Data Matrix
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
Polynomial Regression
Try for 4th degree
polynomial
Other Technology Tools
Excel will also
do regression
Plot data as (x,y) ordered pairs
Right click on data series
Choose Add Trend Line
Other Technology Tools
Use dialog box to
specify regression
Lobster Catch
Millions of Pounds o f
Lobster
60
50
40
30
20
10
0
0
10
20
Years since 1970
30
40
Try Others
Try It Out …
An object is lifted rapidly into the air and then
released. The table shows the height at t
seconds after the start of the experiment
t
0
1
2
3
4
5
6
7
h
0
36
72
108
144
128
80
0
Use your calculator to plot the data
At what time was the object released?
What part of the time interval could be represented
by a linear function? Find that function.
Find a modeling function for the non linear portion.
Assignment
Lesson 4.2B
Page 273
Exercises 41 – 89 EOO