Title of Lesson:
Polynomial Functions of Higher
Degrees
By the end of this lesson, I will be able
to answer the following questions…
1. How do I sketch graphs of polynomial functions using
intercepts, end behavior and strategic points?
2. How do I build functions using intercepts and clues?
3. How do I Build polynomials functions given a realworld scenario and analyze the results using a graphing
calc.
4. What is the Intermediate Value Theorem and what is it
used for?
Vocabulary
1. Multiplicity: Repeated zeros of a function
2. Intermediate Value Theorem: Let a and b be
real numbers such that a < b and f(a)¹ f(b) there
is some value c which is on the interval [a,b] that
guarantees f(a) < f(c) < f (b)
Prerequisite Skills with Practice
Calculator discovery:
Monomials of higher
degrees…
(use different colors)
f (x) = x
2
g(x) = x
4
h(x) = x
6
Properties of
Polynomial graphs
They are always Continuous,
that is – they have no breaks
• are smooth and
They
rounded – no sharp turns
They have predicable end
behavior.
•
End Behavior
f (x) = Ax ...
n
Leading Coefficient Test
•
Leading Coefficient
Test
•
Using the Leading
Coefficient Test.
Describe the end behavior of
the following functions
f(x) = -x + 4x
3
•
• g(x) = 4x 4 + 4x +1
h(x) = f (x)·g(x)
Finding zeros of a
polynomial
function.
f (x) = x 3 - x 2 - 2x
Introducing
multiplicities.
g(x) = -2x 4 + 2x 2
•
Making sketches
based on end
behavior and
intercepts
•
h(x) = x 3 - 4x2 - 25x +100
Find a polynomial
with integer
coefficients given
the following zeros.
2 1
zeros : ,- , 3
5 2
zero : 4
zero : -3
•
• zero : 0, multiplicity : 3
zeros : 3- 2, 3+ 3, 1
Using the
Intermediate Value
Theorem to prove
existence of zeros.
Find three intervals of length
1 in which the polynomial
below is guaranteed to have
a zero.
f (x) = 12x - 32x + 3x + 5
3
•
•
2
A rancher has 374 feet of
fencing to enclose two
adjacent rectangular corrals.
1. Write a function for the
total area with respect to x.
2. Use a graphing calculator
to approximate the dimensions
•that will produce the maximum
Area.
•
Homework:
Page 112: 9,10 (15-43) odd (45-48) all (49-63) odd
•
(79-82) all 91,92 (105-107) all