Section 5.1 Polynomial Functions and Models. Polynomial Functions Three of the families of functions...
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Transcript of Section 5.1 Polynomial Functions and Models. Polynomial Functions Three of the families of functions...
Section 5.1
Polynomial Functions and Models
Polynomial Functions
Three of the families of functions studied thus far:
constant, linear, and quadratic, belong to a much
larger group of functions called polynomials.
We begin our formal study of general polynomials
with a definition and some examples.
Polynomial Functions
A polynomial function is a function of the form
f (x) an xn + an1 xn1 + … + a2 x2 + a1 x + a0
where a0, a1, . . . , an are real numbers and n 1 is
a natural number.
The domain of a polynomial function is ( , ).
Polynomial Functions
Suppose f is the polynomial function
f (x) an xn + an1 xn1 + … + a2 x2 + a1 x + a0
where an 0. We say that, The natural number n is the degree of the polynomial f. The term anxn is the leading term of the polynomial f.
The real number an is the leading coefficient of the polynomial f.
The real number a0 is the constant term of the polynomial f.
If f (x) a0, and a0 0, we say f has degree 0.
If f (x) 0, we say f has no degree.
Determine which of the following functions are
polynomials. For those that are, state the degree.
3 8(a) 3 4f x x x x
(c) 5h x
2 3
(b) 1
xg x
x
(d) ( 3)( 2)F x x x
(a) is a polynomial of degree 8.f (b) is not a polynomial function.
It is the ratio of two distinct polynomials.
g
0
(c) is a polynomial function of degree 0.
It can be written 5 5.
h
h x x 2
(d) is a polynomial function of degree 2.
It can be written ( ) 6.
F
F x x x
Identifying Polynomial Functions
1(e) 3 4G x x x 3 21 2 1(f)
2 3 4H x x x x
(e) is not a polynomial function.
The second term does not have a
nonnegative integer exponent.
G(f) is a polynomial of degree 3.H
Determine which of the following functions are
polynomials. For those that are, state the degree.
Identifying Polynomial Functions
Polynomial Functions: Example
A box with no top is to be built from a 10 inch by
12 inch piece of cardboard by cutting out congruent
squares from each corner of the cardboard and then
folding the resulting tabs.
Let x denote the length of the side of the square
which is removed from each corner.
A diagram representing the situation is,
Polynomial Functions: Example
1. Find the volume V of the box as a function of x. Include an appropriate applied domain.
2. Use a graphing calculator to graph y V (x) on the domain you found in part 1 and approximate the dimensions of the box with maximum volume to two decimal places. What is the maximum volume?
Polynomial Functions: Example
Summary of the Properties of the Graphs of Polynomial Functions
Graphs of Polynomial Functions
Power Functions
A power function of degree n is a function of the
form
f (x) axn
where a 0 is a real number and n 1 is an
integer.
Power Functions: a 1, n even
Power Functions: a 1, n even
Power Functions: a 1, n even
Power Functions: a 1, n odd
Power Functions: a 1, n odd
Power Functions: a 1, n odd
Identifying the Real Zeros of a Polynomial Function and
Their Multiplicity
Graphs of Polynomial Functions
Definition: Real Zero
Find a polynomial of degree 3 whose zeros are
4, 2, and 3.
4 2 3f x a x x x 3 23 10 24a x x x
Finding a Polynomial Function from Its Zeros
The value of the leading coefficient a is, at this point, arbitrary. The next slide shows the graph of three polynomial functions for different values of a.
4 2 3f x x x x
4 2 3f x x x x
2 4 2 3f x x x x
Finding a Polynomial Function from Its Zeros
3 42 2 1 3f x x x x
For the polynomial, list all zeros and their multiplicities.
2 is a zero of multiplicity 1 because the exponent on the factor x – 2 is 1.
1 is a zero of multiplicity 3 because the exponent on the factor x + 1 is 3.
3 is a zero of multiplicity 4 because the exponent on the factor x – 3 is 4.
Definition: Multiplicity
23f x x x
2 2(a) -intercepts: 0 3 0 or 3 0x x x x x
0 or 3x x
2-intercept: 0 0 0 3 0y f 0y
Graphing a Polynomial UsingIts x-Intercepts
23f x x x
0,0 , 3,0
,0 0,3 3,
1
1 16f
Below -axisx
1, 16
1
1 4f
Above -axisx
1,4
4
4 4f
Above -axisx
4,4
23f x x x
x
y
,0 0,3 3,
1
1 16f
Below -axisx
1, 16
1
1 4f
Above -axisx
1,4
4
4 4f
Above -axisx
4,4
Behavior Near a Zero
Example
y = 4(x - 2)
Example
y = 4(x - 2)
Turning Points: Theorem
End Behavior
End Behavior: Example
End Behavior: Example
0 6 so the intercept is 6.f y The degree is 4 so the graph can turn at most 3 times.
4For large values of , end behavior is like (both ends approach )x x
Summary
Analyze the Graph of a Polynomial Function
1The zero has multiplicity 1
2so the graph crosses there.
The zero 3 has multiplicity 2
so the graph touches there.
The polynomial is degree 3 so the graph can turn at most 2 times.
Summary: Analyzing the Graph of a Polynomial Function
The domain and the range of f are the set of all real numbers.
Decreasing: 2.28,0.63
Increasing: , 2.28 and 0.63,