2.2 Polynomial Functions 2015/16 Digital Lesson. HWQ 8/17 Copyright © by Houghton Mifflin Company,...
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Transcript of 2.2 Polynomial Functions 2015/16 Digital Lesson. HWQ 8/17 Copyright © by Houghton Mifflin Company,...
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2.2Polynomial Functions
2015/16
Digital Lesson
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HWQ 8/17
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2
Complete the square on the quadratic
and state the vertex.
2 12 17y x x
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Polynomial Functions are Continuous and Smooth
y
x–2
2y
x–2
2
y
x–2
2
Functions that are not continuous are
not polynomial functions
(Piecewise)
Functions that have sharp turns are not
polynomial functions (Absolute Value)
Polynomial functions have graphs that are
continuous and smooth
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• The polynomial functions that have the simplest graphs are monomials of the form
• If n is even-the graph is similar to
• If n is odd-the graph is similar to
• For n-odd, the greater the value of n, the flatter the graph near(0,0)
y
x–2
2
0,)( nxxf n
2( )f x x
3( )f x x
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Transformations of Monomial Functions
Example 1:
5)( xxf
The degree is odd, the negative coefficient reflects the graph on the x-axis, this graph is similar to
3)( xxf
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Transformations of Monomial FunctionsExample 2:
The degree is even, and has as upward shift of one unit of the graph of
4)( xxf
4( ) ( 1) 1h x x
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A polynomial function is a function of the form1
1 1 0( ) , 0n nn n nf x a x a x a x a a
where n is a nonnegative integer and each ai (i = 0, , n)
is a real number. The polynomial function has a leading coefficient an and degree n.
Examples: Find the leading coefficient and degree of each polynomial function.
Polynomial Function Leading Coefficient Degree5 3( ) 2 3 5 1f x x x x
3 2( ) 6 7f x x x x 5
The Leading Coefficient Test (End Behavior Test)
( ) 1 4f x
2
14
3
0
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• The graph of a polynomial eventually rises or falls. • This can be determined by the function’s degree (odd or even)
and by its leading coefficient (positive or negative)
y
x
–2
2
When the degree is odd:
If the leading coefficient is
positive
The graph falls to the left and rises
to the right
If the leading coefficient is
negative
The graph rises to the
left and falls to the right
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y
x
–2
2
When the degree is even:
If the leading coefficient is
positive
The graph rises to the left and rises
to the right
If the leading coefficient is
negative
The graph falls to the
left and falls to the right
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A real number a is a zero of a function y = f (x)if and only if f (a) = 0.
A polynomial function of degree n has at most n real zeros.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.
1. x = a is a zero of f.
2. x = a is a solution of the polynomial equation f (x) = 0.
3. (x – a) is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
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Zeros of Polynomial Functions
The graph of f has at most n-1 relative extrema (relative minima or maxima.)
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0,)( nxxf n
A polynomial function of degree n has at most n real zeros. It has exactly n total zeros (both real and imaginary.)
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Use the leading coefficient test (end behavior test)to describe left and right hand behavior and sketch the graph
Example 1
xxxf 4)( 3
y
x
–2
2
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Use the leading coefficient test (end behavior test)to describe left and right hand behavior and sketch the graph
Example 2
45)( 24 xxxf
y
x
–2
2
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Use the leading coefficient test (end behavior test)to describe left and right hand behavior and sketch the graph
Example 3
xxxf 5)(
y
x
–2
2
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y
x–2
2
Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2.
Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).
The real zeros are x = –1, x = 0, and x = 2.
Notice that there is a zero at x = 0 that has a multiplicity of 2. A zero with an even multiplicity will bounce off the x-axis.
f (x) = x4 – x3 – 2x2
(–1, 0) (0, 0)
(2, 0)
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Finding Zeros of a Polynomial Function
• Student ExampleFind all real zeros of
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xxxxf 2)( 23
The zeros are 0, 2, 1x x x
The x-intercepts are 0,0 , 2,0 , 1,0
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Example continued:Sketching the graph of a Polynomial Function with known zeros:
Sketch a graph by hand.
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y
x
–2
2
xxxxf 2)( 23
The x-intercepts are 0,0 , 2,0 , 1,0
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Finding a Polynomial Function with Given Zeros
Write an equation for a polynomial function with zeros at x = -2, 1, and 3. Sketch a graph by hand.
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y
x
–2
2
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Finding a Polynomial Function with Given Zeros
Student Example: Find a polynomial function
with the given zeros: x = -1, 2, 2
Sketch a graph by hand.
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y
x
–2
2
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Sketching the Graph of a Polynomial Function
Sketch the graph of
1. What is the end behavior?
2. Find the zeros of the polynomial function.
3. Plot a few additional points.
4. Draw the graph.
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3 2( ) 3 9f x x x
y
x
–2
2
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Sketching the Graph of a Polynomial Function
Sketch the graph of
1.What is the end behavior?
2.Find the zeros.
3.Plot a few additional points.
4.Draw the graph.
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343)(4
xxxf
y
x
–2
2
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Homework
• Section 2.2 pg. 108
1-7 odd, 17-43 odd, 49-55 odd, 61
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