Polynomial Functions of Higher Degree p' 275283

1
Polynomial Functions of Higher Degree p. 275-283

• OBJECTIVES
• Use transformations to sketch graphs of
  polynomial functions
• Determine the end behavior of graphs of
  polynomial functions
• Use zeros of polynomial functions as sketching
  aids

2
Graphs of Polynomial Functions p. 275

• Features of polynomial functions
• Continuous graphs
• (No breaks or discontinuities)

• Smooth
• (No sharp corners)

3

• Consider the equation y x3
• What is the degree?
• What is the leading coefficient?

n 3 1
y
x y
(x, y)
2
8
(2, 8 )
1
1
(1, 1)
0
0
(0, 0)
1
x
1
(1, 1)
2
8
(2, 8)
What happens to y as x increases?
as
4
Leading Coefficient Test p. 277For the
polynomial function f(x) anxn a1x
a0

• 1. When n is odd and an gt 0

as
as

• 2. When n is odd and an lt 0

as
as
5

• Consider the equation y x4 x
• What is the degree?
• What is the leading coefficient?

n 4 1
x y
(x, y)
2
18
(2, 18 )
y
1
2
(1, 2)
0
0
(0, 0)
0
x
1
(1, 0)
2
14
(2, 14)
What happens to y as x increases?
as
6
p. 277 For the polynomial function f(x)
anxn a1x a0
1. When n is even a gt 0
as
as
2. When n is even and an lt 0
as
as
7
Real Zeros of Polynomial Functions p. 279

• If f is a polynomial and a is a real number, the
  following statements are equivalent
• 1. x a is a zero of the function f.
• 2. x a is a solution of the 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 f.

8
Fascinating Facts p. 278-279

• The graph of f has, at most, n-1 turning
  points.
• The function has, at most, n real zeroes.
• A factor (x a)k, k gt 1, yields a repeated zero
  x a of multiplicity k.
• If k is odd, the graph crosses the x-axis at x
  a.
• If k is even, the graph touches the x-axis at x
  a.

9
Homework

• P. 284-288 1-42, 47-80 alt odd, 89-92 odd
• Synthesis 99-103
• Review 105-122
• Read p. 289-294 (3.3)
• PRE QUIZ 3.3
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