October 07, 2015
(2.3) Polynomial Functions
Obj: To learn about Polynomial functions, their End Behavior, and
how to find the Zeros.
Why: These topics are important in modeling (volume of a box), and
can be used to provide approximations to more complicated
functions studied in calculus.
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Polynomial Functions:
f(x) = axn + bxn-1 + cxn-2 + ... + gx2 + hx + k = 0
leading term coefficient
defined and continous along (-
deg. = n
constant term, f(0)= k
y-int.
,
)
The graph of a polynomial function is continuous, has no breaks, holes
or gaps, is smooth, rounded turns, not sharp turns.
End Behavior of Polynomial Functions when:
a. the degree is even
b. the degree is odd
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Graph.
1. f(x) = -(x+5)
3
2. g(x) = 3(x-1)4 - 2
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Graph the function, locate any extrema and zeros, and
explain how it is related to the monomials from which is
is built.
f(x) = x3 - x
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
A polynomial function of degree n has at most n-1 local
extrema and at most n zeros.
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Graph f(x) = x3 + 2x2 -11x -12 in a window showing its
extrema and zeros and its end behavior. Describe the
end behavior using limits.
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Find the zeros of:
f(x) = x3 - x2 - 6x
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Find the zeros of f(x) = (x-2)3(x+1)2
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Zeros of Odd and Even Multiplicity
If a polynomial function has a real zero of odd multiplicity,
then the graph of f crosses the x-axis at that zero.
If a polynomial function has a real zero of even multiplicity,
then the graph of f touches the x-axis at that zero.
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
Find all real zeros of
f(x) = x4 + 0.1x3 -6.5x2 + 7.9x - 2.4
October 07, 2015
Obj: To learn about Polynomial functions, their end behavior, and
how to find the zeros.
HW
(HR) (2.3) Pg.193: 5, 9-12, 17, 19, 21, 29-32, 33-41odd, 53, 61, 67,
73