Transcript 2 2 Polynomial Functions

Objectives:
1. To find the degree,
zeros, number of
turning points, and
the end behavior of
polynomial graphs
2. To sketch the graphs
of polynomial
functions
Assignment:
• P. 148: 1-8 (Some)
• P. 149: 13-22 (Some)
• P. 149: 27-42 (Some)
– Sketch graphs
•
•
•
•
P. 149: 57-66 (Some)
P. 149: 67-80 (Some)
P. 152: 99-102
Read: P. 153-158
Polynomial
Coefficient
Leading Coefficient
Constant Term
Degree
Linear
Quadratic
Cubic
Quartic
End Behavior
Turning Points
Multiplicity
You will be able to find
the degree of
polynomials
A polynomial function in x is a function of the
form:
n
2
f ( x)  an x 
 a2 x  a1 x  a0
• Exponents:
– All exponents are nonnegative integers
– The degree of the polynomial is n, as long as
an  0
– The degree is the highest power of the polynomial
A polynomial function in x is a function of the
form:
n
2
f ( x)  an x 
 a2 x  a1 x  a0
• Coefficients:
– All coefficients ak are real numbers
– an is the leading coefficient
– a0 is the constant term
Degree
Type
Polynomial Function
0
Constant
f(x) = a0
1
Linear
f(x) = a1x + a0
2
Quadratic
f(x) = a2x2 + a1x + a0
3
Cubic
f(x) = a3x3 + a2x2 + a1x + a0
4
Quartic
f(x) = a4x4 + a3x3 + a2x2 + a1x + a0
5
Quintic
f(x) = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
6
Sextic
f(x) = a6x6 + a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0
Degree
Type
Examples
0
Constant
f(x) = 42
1
Linear
f(x) = 3x – 7
2
Quadratic
f(x) = -1.5x2 + .75x + 6
3
Cubic
f(x) = x3 + 4x2 – 5x + 1
4
Quartic
f(x) = 7x4 + 9.5x3 +2x2 + 10x – 1
5
Quintic
f(x) = 4x5 + 8x4 + 15x3 + 16x2 + 23x + 42
6
Sextic
f(x) = x6 – 2x5 + 3x4 – 4x3 + 5x2 – 6x + 7
You will be able to
determine the
end behavior of
polynomial
graphs
The graphs of polynomial functions are
continuous; that is, they have no holes or
gaps.
The graphs of polynomial functions are smooth
curves; that is, they have no sharp turns.
In this demonstration, we
will investigate the
look, feel, and taste of
the “end” behavior of
polynomial graphs.
(“End” is in quotation
marks because, strictly
speaking, most graphs
don’t have “ends”.)
End behavior refers to
what a graph looks like
as x approaches
positive infinity
x  
or as x approaches
negative infinity
x  
Now look at all the functions of even degree. What do
you notice about their end behavior? What about
the end behavior of odd functions?
Just so you know, your book calls end behavior lefthand and right-hand behavior. Just wanted to avoid
any confusion.
End Behavior: Even and Odd
Even End Behavior
Odd End Behavior
Describe the degree and
leading coefficient of
the polynomial whose
graph is shown.
You will be able to find the zeros
of a polynomial and determine the
behavior of the graph at those
points
A consequent of the
Fundamental Theorem of
Algebra and it’s Corollary is
that:
• An nth degree polynomial
has n zeros.
• Sometimes at least one of
these zeros repeats k times
and is said to be a repeated
root with a multiplicity of k.
y  x 2 ( x  3)( x  5)3
x0
x0
x3
x  5
x  5
x  5
6 total zeros, but only 3
x-intercepts (some repeat)
1. How many solutions does the equation
x4 + 8x2 – 5x + 2 = 0 have?
2. How many zeros does the function
f(x) = x3 + x2 – 3x – 3 have?
1. How many zeros does a quintic polynomial
have?
2. How many x-intercepts does a quintic
polynomial have?
3. Why are the answers above not necessarily
the same?
Determine the degree of the following polynomial
functions. How many zeros does each have? Use a
graphing calculator to determine how the multiplicity
of each zero affects the graph.
1. f(x) = (x + 5)(x – 1)2
2. g(x) = (x + 5)3(x – 1)4
3. h(x) = (x + 5)5(x – 1)6
Real Zeros:
• Only real zeros are xintercepts. Imaginary zeros do
not touch the x-axis.
Odd Multiplicity:
• A zero of odd multiplicity
crosses the x-axis at that zero.
Even Multiplicity:
• A zero of even multiplicity is
tangent to the x-axis at that
zero.
Odd
Multiplicity
Even
Multiplicity
 x  1
 x  2
3
2
And here’s one more important tidbit about the
graphs of polynomials:
Let f be a polynomial function of degree n.
• Then f has at most n – 1 turning points
• If f has n distinct real roots, then f has exactly
n – 1 turning points
1. The graph of a polynomial function has 4 xintercepts. What is the smallest possible
degree of this function?
2. The graph of a polynomial function has 4
turning points. What is the smallest possible
degree of the function?
1. The graph of a polynomial function has k xintercepts. What is the smallest possible
degree of this function?
2. The graph of a polynomial function has k
turning points. What is the smallest possible
degree of the function?
Find a polynomial of the indicated degree that
has the zeros x = 0, 1, -1, -6.
1. Degree n = 4
2. Degree n = 5
Find a polynomial function with the indicated
number of turning points that has zeros x = 2,
-5, 3, 7.
1. Turning points = 3
2. Turning points = 4
If f is a polynomial
function of degree n,
then its graph only
does really interesting
things near and
between its zeros.
– This is where all the
turny bits happen.
To graph a polynomial function:
1. Find and plot the intercepts (x- and y-)
2. Determine the end behavior
3. Plot points near and between zeros (relative
or absolute min/max)
4. Connect the points with a reasonable curve
keeping in mind the end behavior
Graph each of the following polynomial
functions the old-fashioned way (in other
words, by hand).
1. f(x) = 0.25(x + 2)(x – 1)(x – 3)
2. g(x) = -2(x – 1)2(x – 4)
3. h(x) = x3(x – 2)(x + 3)
Graph each of the following polynomial
functions the old-fashioned way (in other
words, by hand).
1. f(x) = x4 – 12x3 + 36x2
2. g(x) = 3x5 + 13x4 – 10x3
3. h(x) = -x3 + 4x2 +25x – 100
Objectives:
1. To find the degree,
zeros, number of
turning points, and
the end behavior of
polynomial graphs
2. To sketch the graphs
of polynomial
functions
Assignment:
• P. 148: 1-8 (Some)
• P. 149: 13-22 (Some)
• P. 149: 27-42 (Some)
– Sketch graphs
•
•
•
•
P. 149: 57-66 (Some)
P. 149: 67-80 (Some)
P. 152: 99-102
Read: P. 153-158