3.5 Higher – Degree Polynomial
Functions and Graphs
Polynomial Function
Definition: A polynomial function of degree
n in the variable x is a function defined by
P(x) = anxn + an-1xn-1 + … + a1x + a0
Where each ai(0 ≤ i ≤ n-1) is a real number,
an ≠ 0, and n is a whole number. What’s
the domain of a polynomial function?
Get to know a polynomial function
P(x) = anxn + an-1xn-1 + … + a1x + a0
an : Leading coefficient
anxn : Dominating term
a0 : Constant term
Cubic Functions
P(x) = ax3 + bx2 + cx + d
(a)
(c)
(b)
(d)
Quartic Functions
P(x) = ax4 + bx3 + cx2 + dx + e
(a)
(c)
(b)
(d)
Extrema




Turning points: points where the function
changes from increasing to decreasing or
vice versa.
Local maximum point: the highest point at a
peak. The corresponding function values are
called local maxima.
Local minimum point: the lowest point at a
valley. The corresponding function values are
called local minima.
Extrema: either local maxima or local
minima.
Absolute and Local Extrema

Let c be in the domain of P. Then
(a) P(c) is an absolute maximum if P(c)
≥ P(x) for all x in the domain of P.
(b) P(c) is an absolute minimum if P(c)
≤ P(x) for all x in the domain of P.
(c) P(c) is an local maximum if P(c) ≥
P(x) when x is near c.
(d) P(c) is an local minimum if P(c) ≤
P(x) when x is near c.
Example
Local minimum
point
Local minimum
point
Local minimum
point
A function can only
have one and only
one absolute
minimum of
maximum
Local minimum
point
Local minimum &
Absolute minimum
point
Hidden behavior
Hidden behavior of a polynomial function is
the function behaviors which are not
apparent in a particular window of the
calculator.
Number of Turning Points

The number of turning points of the
graph of a polynomial function of degree
n ≥ 1 is at most n – 1.

Example: f(x) = x
f(x) = x2
f(x) = x3
End Behavior

Definition: The end behavior of a
polynomial function is the increasing of
decreasing property of the function
when its independent variable reaches to
∞ or - ∞

The end behavior of the graph of a
polynomial function is determined by the
sign of the leading coefficient and the
parity of the degree.
End Behavior
a>0
Odd degree
a<0
a>0
Even degree
a<0
example

Determining end behavior Given the
Polynomial
f(x) = x4 –x2 +5x -4
X – Intercepts (Real Zeros)

Theorem: The graph of a polynomial
function of degree n will have at most n
x-intercepts (real zeros).

Example: P(x) = x3 + 5x2 +5x -2
Comprehensive Graphs

A comprehensive graph of a polynomial
function will exhibit the following
features:
1. all x-intercept (if any)
2. the y-intercept
3. all extreme points(if any)
4. enough of the graph to reveal the
correct end behavior
example


1. f(x) = 2x3 – x2 -2
2. f(x) = -2x3 - 14x2 + 2x + 84
a) what is the degree?
b) Describe the end behavior of the
graph.
c) What is the y-intercept?
d) Find any local/absolute maximum
value(s). ... local/absolute maximum points.
[repeat for minimums]
e) Approximate any values of x for which
f(x) = 0
Homework

PG. 210: 10-50(M5), 60, 63

KEY: 25, 60

Reading: 3.6 Polynomial Fncs (I)