Transcript Polynomial Functions and Models Lesson 4.2 Review  General polynomial formula P( x)  an x  an1x n     n1  ...

Polynomial Functions
and Models
Lesson 4.2
Review
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General polynomial formula
P( x)  an x  an1x
n
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n1
 ...  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
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Turning point
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A point (x, y) on the graph
Located where graph changes
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from increasing
to decreasing
(or vice versa)
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Family of Polynomials
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Constant polynomial functions
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Linear polynomial functions
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f(x) = a
f(x) = m x + b
Quadratic polynomial functions
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f(x) = a x2 + b x + c
Family of Polynomials
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Cubic polynomial functions
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f(x) = a x3 + b x2 + c x + d
Degree 3 polynomial
Quartic polynomial functions
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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
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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
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From this view, they appear very similar
Contrast Short Run Behavior
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Now Change the window to be
-5 < x < 5 and -35 < y < 15
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How do the functions appear to be different from
this view?
Contrast Short Run Behavior
Differences?
 Real zeros
 Local extrema
 Complex zeros
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Note: The standard form of the polynomials does not
give any clues as to this short run behavior of the
polynomials:
Assignment
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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

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Consider
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Sketch graph
Use calculator to display
Consider if f(x) is continuous
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Linear Regression
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Used in section previous lessons to find
equation for a line of best fit
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Other types of
regression are
available
Polynomial Regression
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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
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Specify the plot F2,
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X's from C1, Y's from C2
View the
graph
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Check Y= screen, use Zoom-Data
Polynomial Regression
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Try for 4th degree
polynomial
Other Technology Tools
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Excel will also
do regression
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Plot data as (x,y) ordered pairs
Right click on data series
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Choose Add Trend Line
Other Technology Tools
Use dialog box to
specify regression
Lobster Catch
Millions of Pounds o f
Lobster
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60
50
40
30
20
10
0
0
10
20
Years since 1970
30
40
Try Others
Try It Out …
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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
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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
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Lesson 4.2B
Page 273
Exercises 41 – 89 EOO