DAY 14
Warm – up Graph the following quadratic
1. y = (x – 3)(x + 1)
3. y = (x + 2)2
2. y = (x – 2)(x + 2)
Notes Graphing Polynomials
Highest Degree is ODD
Leading
coefficient is
positive
Leading
coefficient is
negative
Highest Degree is EVEN
As x  – ∞,
As x  – ∞,
As x  ∞,
As x  ∞,
As x  – ∞,
As x  – ∞,
As x  ∞,
As x  ∞,
End Behavior Patterns:
Graph each polynomial function in the table. Then complete the table to describe the end behavior of the graph of
each function.
Degree of
Polynomial
Function
Sign of
Lead
Coefficient
As
As
What
will
a
graph
with
the
following
characteristics
look
like?
a)
a
degree
of
five
and
a
negative
leading
coefficient.
b) a
degree
of
six
and
a
positive
leading
coefficient.
Lead
Coefficient
#
of
Zeros
#
of
Turns
Degree
Multiplicity
Lead
Coefficient
Polynomial
y
=
2(x
+
2)2(x
–
4)
#
of
Zeros
Degree
Actual
Zeros
and
their
Multiplicity
#
of
Turns
y
=
‐x(x
–
7)(x
+
4)
y
=
(x2
+
1)(x
–
1)
y
=
x(x
+
3)2(x
+
1)
y
=
‐(x2
–
4)(x2
–
9)
y
=
‐x(x
–
2)(x2+3)
Given
the
polynomial
y
=
‐2(x
+
1)2(x
–
2)(x
–
3)3(x2
+
2)
determine
the
following
without
graphing.
a)
Determine
the
end
behavior
of
the
graph.
b)
Determine
the
zeros
of
the
function.
c)
Determine
the
x‐intercepts
of
the
function.
d)
Determine
the
y‐intercept
of
the
function.
e)
Describe
the
behavior
of
the
graph
at
each
of
the
x‐intercepts.
f)
Without
using
a
calculator,
sketch
the
graph
of
the
function
Polynomial graphing summary:
Standard Form
Factored Form
Equation
Degree
the highest power of x
write out ALL of the factors(there
should be NO exponents left
outside the parentheses) and then
add up the exponents on the x’s
Lead
Coefficient
the number in front of the x with the highest
exponent
after writing out all of the factors,
multiply the coefficients of all of
the x’s and a(the number in front
of the entire expression)
End
Behavior
+ lead
-lead
even
degree
up/up
down/down
odd
degree
down/up
up/down
SAME
Zeros (xintercepts)
factor and then set each factor equal to zero
set each factor equal to zero
Multiplicity
the number of times each zero occurs
the exponent of the factor that
your zero comes from in the
original equation
Behavior at
zeros
if multiplicity is even touches
SAME
y-intercept
set x equal to zero and solve for y
if multiplicity is odd crosses
SAME
Standard Form
Equation
Degree
Lead
Coefficient
End
Behavior
Zeros (xintercepts)
Multiplicity
Behavior at
zeros
y-intercept
Table of
extra values
for graphing
Factored Form
EXAMPLE ONE Graphing a polynomial from factored form
1
1. y = –(x – 1)(x + 2)(x + 3)
2. y = x + 4) (x + 2) (x – 1)2
2
YOU TRY Graphing a polynomial from factored form
1
1. y = (x + 2)2 (x – 2)2
4
2. y = (x + 3)(x + 2) (x – 1)
EXAMPLE TWO Tell me EVERYTHING you can about this function!
y = –x3 – x2 + 6x + 6
1. f(x) = x4 – 9x2 + 4x + 12
Zeros:
Zeros:
Y intercepts:
Y intercepts:
Maximum:
Maximum:
Minimum:
Minimum:
Increasing:
Increasing:
Decreasing:
Decreasing:
PRACTICE
1.
The
graph
of
the
polynomial
function
is
shown.
What
is
true
about
the
function’s
degree
and
leading
coefficient?
a)
The
degree
is
odd
and
the
leading
coefficient
is
positive.
b)
The
degree
is
odd
and
the
leading
coefficient
is
negative.
c)
The
degree
is
even
and
the
leading
coefficient
is
positive.
d)
The
degree
is
even
and
the
leading
coefficient
is
negative.
2.
Describe
the
degree
and
leading
coefficient
of
the
polynomial
function
shown.
a.
b.
c.
Describe
the
end
behavior
of
the
graph
of
the
polynomial
function
by
completing
the
blank:
3.
f (x ) = 10x 4 4.
f (x ) = −x 6 + 4x 3 − 3x f (x )→ ____, as x → −∞ f (x )→ ____, as x → −∞ f (x )→ ____, as x → ∞ f (x )→ ____, as x → ∞ 5.
f (x ) = −2x 3 + 7x − 4 6.
f (x ) = x 7 + 3x 4 − x 2 f (x )→ ____, as x → −∞ f (x )→ ____, as x → −∞ f (x )→ ____, as x → ∞ f (x )→ ____, as x → ∞ 7.
f (x ) = 3x 10 − 16x 8.
f (x ) = −6x 5 + 14x 2 + 20 f (x )→ ____, as x → −∞ f (x )→ ____, as x → −∞ f (x )→ ____, as x → ∞ f (x )→ ____, as x → ∞ 9.
Explain
what
a
local
maximum
of
a
function
is
and
how
it
may
be
different
from
the
maximum
value
of
the
function.
Use
a
graphing
calculator
to
graph
the
polynomial
function.
Identify
the
x‐intercepts
and
the
points
where
the
local
maximums
and
local
minimums
occur.
22.
f (x ) = 2x 3 + 8x 2 − 3 23.
f (x ) = 0.5x 3 − 2x + 2.5 f (x ) = −x 4 + 3x 25.
f (x ) = x 5 − 4x 3 + x 2 + 2
24.
Graph
the
functions
(without
a
calculator).
10.
f (x ) = (x + 1)2 (x − 1)(x − 3) 11.
f (x ) = 4(x + 1)(x + 2)(x − 1) 13.
f (x ) = (x − 4)(2x 2 − 2x + 1) 12.
f (x ) = 2(x + 2)2 (x + 4)