sections 9.2 and 9.3 polynomial functions. what is a power function? what happens if we add or...
TRANSCRIPT
![Page 1: Sections 9.2 and 9.3 Polynomial Functions. What is a power function? What happens if we add or subtract power functions? A polynomial is a sum (or difference)](https://reader033.vdocuments.mx/reader033/viewer/2022042718/56649edc5503460f94bec07c/html5/thumbnails/1.jpg)
Sections 9.2 and 9.3Polynomial Functions
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• What is a power function?
• What happens if we add or subtract power functions?
• A polynomial is a sum (or difference) of power functions whose exponents are nonnegative integers
• What determines the degree of a polynomial?
• For example
• What is the leading term in this polynomial?
1103 2 xxy
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• Which of the following are polynomials (and what is their degree)?
5)(.4
13)(.3
2)(.2
10023)(.13
354
xk
xxxh
exxxg
xxxxxf
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• What are the zeros (or roots) of a polynomial?– Where the graph hits the x axis– The input(s) that make the polynomial equal to 0
• How can we find zeros of a polynomial?
• For example, what are the zeros of
• Notice this polynomial is in its factored form– It is written as a product of its linear factors
)5)(3()( xxxh
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Polynomials
32
2
3
2
)3()(.5
)3()(.4
)2)(1()(.3
)4)(1)(2()(.2
)5)(3()(.1
xxxn
xxxm
xxxk
xxxxg
xxxf
• Determine the degree and the zeros of the following polynomials?
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Behavior of Polynomials
• Using your calculator, graph the following functions and compare the graphs
• What do you notice about the behavior of the graph at the zeros for m(x) and n(x)?
32
2
)3()(
)3()(
xxxn
xxxm
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Behavior of Polynomials
)3()( 2 xxxm 32 )3()( xxxn
What behavior do you notice at the zeros of these functions?
xx
What is the significance of this point?
What is the significance of this point?
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Multiplicity of Roots/Zeros• When a polynomial, p, has a repeated
linear factor, then it has a multiple root.– If the factor (x - k) is repeated an even number
of times, the graph does not cross the x-axis at x = k. It ‘bounces’ off. – Note that the concavity does not change at x = k
– If the factor (x - k) is repeated an odd number of times, the graph crosses the x-axis, but flattens out at x = k. – Note that we will have an inflection point at x = k
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Behavior of Polynomials
• Consider the function:
• Complete the tables:
3)( xxf
x f(x)
2
10
100
x f(x)
-2
-10
-100
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Behavior of Polynomials
• Consider the function:
• Complete the tables:
• What can you say about f(x) as x ∞?• What can you say about f(x) as x -∞?
3)( xxf
x f(x)
2 8
10 1,000
100 1,000,000
x f(x)
-2 -8
-10 -1,000
-100 -1,000,000
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Limit Notation:
• Another way to notate long-run or end-behavior of functions is by using “limit notation.” – We can notate “the limit of f(x) as x goes to infinity” by
writing:
The above expression signals you to evaluate what the output value of the function f approaches as x gets larger and larger.
– We can notate “the limit of f(x) as x goes to negative infinity” by writing:
lim ( )x
f x
lim ( )x
f x
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End Behavior
• Consider the following two functions
4
24
)(
2065)(
xxg
xxxxf
x f(x) g(x) f(x)/g(x) % change
2
10
100
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End Behavior
• Consider the following two functions
4
24
)(
2065)(
xxg
xxxxf
x f(x) g(x) f(x)/g(x) % change
2 68 16 4.25 325%
10 10,580 10,000 1.058 5.8%
100 10,050,620 10,000,000 1.005062 0.506%
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End BehaviorConsider the graphs following two functions
2065)( 24 xxxxf 4)( xxg
Let’s see what happens as we zoom out
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2065)( 24 xxxxf 4)( xxg
Let’s see what happens as we zoom out some more
End BehaviorConsider the graphs following two functions
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2065)( 24 xxxxf 4)( xxg
Let’s see what happens as we zoom out some more
End BehaviorConsider the graphs following two functions
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End BehaviorConsider the graphs following two functions
2065)( 24 xxxxf 4)( xxg
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End Behavior
• Consider the following two functions
• Find the following:
• A functions end behavior is determined by its leading term
4
24
)(
2065)(
xxg
xxxxf
)(lim)(lim
)(lim)(lim
xgxg
xfxf
xx
xx
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End Behavior• Both end behavior and degree are determined by
the lead term• Is there any relationship between the degree of a
polynomial and its end behavior?
2( )f x x3( )f x x
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• Find a possible polynomial for the following graph
– Is it the only possibility?
– What is the minimum possible degree?