determine the following algebraically (no calculator) a)vertex b)x-intercepts c)y- intercepts d)is...
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Determine the following algebraically (no calculator)
a) Vertex
b) x-intercepts
c) y- intercepts
d) Is the vertex a max or min? How do you know without
graphing?
1123)( 1. 2 xxxfQuadratics
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Section 3.2 Polynomial Functions and Models
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1. Polynomial Functions and their degree
Find the degree (largest power when written in standard form).
Factored form: Add the degrees contributed by each factor523 )2()2()( :2 xxxxfExample
32)( :1 6 xxxfExample
f (x) anxn an 1x
n 1 ... a1x a0
A polynomial function is a function of the form
Coefficients are: n is non-negative integer.
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2. Properties of the Power Functions
A power function is of the form
f (x) x n
If n is odd integer If n is even integer
Symmetry:
Domain:
Range:
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3. End Behavior of Polynomials
Given a polynomialAs x increases/decreases without bound, the highest degree term determines the end behavior of f(x).
01)( axaxaxaxf nn
nn
nnxa
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3. End Behavior of the graph
Even Degree and
Leading Term Testlook at leading term (from standard form)
nnxa
Leading Coefficient is Positive
,..., 42 xx
Leading coeffic.
is Negative,..., 42 xx
Odd Degree and
Leading coefficient is Positive
,..., 53 xx
Leading coefficient is Negative
,..., 53 xx
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3b) Determine End Behavior using Leading Coefficient Test
1)
2)
3)
112)( xxxf
nnxa
xxxf 23)(
322 )2()2(2.0)( xxxxf
p. 183 #23-34. What is the end behavior of these graphs?
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3c) End Behavior can also be used to determine Window Size on Calculator
248)( 234 xxxxf
How do you know if you are viewing the
entire graph?
Set window to [-5,5] x [-5,5] and graph:
41013)( 23 xxxxf
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4. Zeroes of Polynomials and their Multiplicity
Factor Theorem If is a polynomial function, for which
the following are equivalent statements:
(1) r is a zero or root of
(2) r is an x-intercept of
(3) is a factor of
f
f
f
f)( rx
0)( rf
Example 1: Determine all zeros of f.82)( 2 xxxf
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4 Zeroes of Polynomials and their MultiplicityDefinition: The multiplicity of a zero is the degree of the factor
Example: If f has a factor , then 3 is a zero with multiplicity 4
4)3( x
Example:
Determine: The zeros of f and multiplicity of each :
)2()3(2)( 42 xxxxf
Multiplicity greater than 1 represents a repeated zero
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5. Investigating the Role of Multiplicity.How does Multiplicity affect the behavior of the graph at the zero?
Graph each function, what do you notice changes at the zero when the degree is even or odd?
1) 3)
2) 4)
Does the sign of change on each side of the zero?)(xf
3)3()( xxf3)4()( xxf
2)3()( xxf2)4()( xxf
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6. Behavior of Graph at a ZeroMultiplicity tells us the behavior of the graph at a zero (x-intercept).
If multiplicity , m, is a number that is:
Behavior of graph at the zero
Even touches (is tangent)Odd crosses (changes sign)
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6. Example Continued
Zeros Multiplicity Is m an even or odd integer?
Behavior of graph at the zero
0 2 3 4 -2 1
Previous Example . )2()3(2)( 42 xxxxf
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6. More Practice
342 )1()1()( xxxxf
State the degree of this polynomial.How many zeros does this function have?
Zeros of the function
Multiplicity of the zero
Is meven or odd?
Shape of the graph near the zero
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Use your graphing calculator to graph the following:1) 2)2()( xxf 6)2()( xxf
7. Large values of Multiplicity
Analyze the graph of
What happens at the zero as m gets large?
mxxf )2()(
2)3)2()( xxf 7)2()( xxf
The graph flattens out at the zero as the multiplicity increases.
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8. Graphing a Polynomial Function
Given the polynomial
1. Leading Term: 2. End behavior:
3. y-intercept:
4. x-intercepts (i.e. the zeros)
23 )3)(2()1()( xxxxf
Zeros Multiplicity even or odd? Cross/Touch
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8. Graphing a Polynomial Function
Given the polynomial
Before finding zeros: Write in completely factored form1. Leading Term:2. End behavior
3. y-intercept4. x-intercepts (i.e. the zeros)
)4(5.)( 23 xxxf
Zeros Multiplicity even or odd? Cross/Touch
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8. Graphing a Polynomial Function
Your turn...
1. Degree2. End behavior
3. y-intercept4. x-intercepts (i.e. the zeros)
52 )5)(1()1()( xxxxxf
Zeros Multiplicity even or odd? Cross/Touch
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1. End behavior
2. y-intercept
3. x-intercepts (i.e. the zeros)
8. Graphing a Polynomial Function
When polynomial is not in factored form, find zeros using:
Factoring or Graphing Calculator.
9182)( 23 xxxxf
Zeros Multiplicity even or odd? Cross/Touch
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10. Using the graphing calculator
Graph: p. 184 #81.
x-intercepts: Use ZERO feature y-intercepts: TRACE: x=0
d) Table to determine graph close to zero. Is it above or below?
e) Max/Min
Find zeros (x-intercepts) using graphing calculator.
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9. a) Building Polynomials from zerosGiven that f has zeros with multiplicitywe can write:
Write one possible polynomial with these properties:1) Degree 4: Zeroes at: -1, 2, 5 ;
21 )()()( 21mm rxrxaxf
,, 21 rr ,, 21 mm
2) x-intercepts at: (-3,0), (4,0), (-1,0), (0,0) and the graph rises to the left and falls to the right.
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Polynomial – turning points
Definition: turning point - graph changes between increasing and decreasing.
If polynomial is degree n
Then it will have AT MOST n-1 turning points.
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9. a)Building Polynomials from a graph
Use y-intercept to find scale factor.
Zerosx-intercepts
C/T ? Multiplicity
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More practice….Construct a polynomial function that might have this graph.
Use higher-degree if graph is “flat” at the zero
Zerosx-intercepts
C/T ? Multiplicity
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11. Finding a function of best fitUsing the graphing calculator, we can find a function of best fit for the following relationships:
Quadratic :cbxaxy 2 Cubic:
dcxbxaxy 23
Count the # of turning points
to determine best function
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9. a)Building Polynomials from a graph
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12. Analyzing the Graph of a Polynomial Function
Given the polynomial
1. End behavior.
2. x-intercepts. Solve f(x) = 0
)1)(3()4(2)( 22 xxxxxf
3. y-intercepts. Find f(0).
a) Behavior at each intercept (even/odd)
b) If k > 1, graph flattens for larger values of k.
5. Turning points: Graph changes between increasing/decreasing.
4. Symmetry: Odd/Even