sections 9.2 and 9.3 polynomial functions. what is a power function? what happens if we add or...
TRANSCRIPT
Sections 9.2 and 9.3Polynomial Functions
• 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
• Which of the following are polynomials (and what is their degree)?
5)(.4
13)(.3
2)(.2
10023)(.13
354
xk
xxxh
exxxg
xxxxxf
• 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
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?
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
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?
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
Behavior of Polynomials
• Consider the function:
• Complete the tables:
3)( xxf
x f(x)
2
10
100
x f(x)
-2
-10
-100
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
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
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
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%
End BehaviorConsider the graphs following two functions
2065)( 24 xxxxf 4)( xxg
Let’s see what happens as we zoom out
2065)( 24 xxxxf 4)( xxg
Let’s see what happens as we zoom out some more
End BehaviorConsider the graphs following two functions
2065)( 24 xxxxf 4)( xxg
Let’s see what happens as we zoom out some more
End BehaviorConsider the graphs following two functions
End BehaviorConsider the graphs following two functions
2065)( 24 xxxxf 4)( xxg
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
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
• Find a possible polynomial for the following graph
– Is it the only possibility?
– What is the minimum possible degree?