Download polynomial function

How do I analyze a polynomial
function?
Daily Questions:
1) What is polynomial function?
2)How do I determine end behavior?
EVALUATING POLYNOMIAL FUNCTIONS
A polynomial function is a function of the form
f(x) = an x nn + an – 1 x nn – 11 +· · ·+ a 1 x + aa00
Where ann  00 and the exponents are all whole numbers.
For this polynomial function, aan is the leading coefficient,
coefficient
n
aa00 is the constant
constant term,
term and n is the degree.
degree
A polynomial function is in standard form if its terms are
descending order
order of
of exponents
exponents from
from left
left to
to right.
right.
written in descending
Examples of Polynomial Functions
3x  2 x  3x  7 x  2 x  9
3
2
12 x  6 x  4 x  1
5
4
3
2
 7 x  2 x  8x  2 x  3
4
3
2
What do you notice about all these equations?
All exponents must be whole numbers and
coefficients are all real numbers…
Graphs of polynomial functions are continuous. That is, they
have no breaks, holes, or gaps.
f (x) = x3 – 5x2 + 4x + 4
y
y
x
continuous
smooth
polynomial
y
x
not continuous
not polynomial
x
continuous
not smooth
not polynomial
Polynomial functions are also smooth with rounded turns. Graphs
with points or cusps are not graphs of polynomial functions.
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
f (x) = x 3 + 3 x
SOLUTION
The function is not a polynomial function because the
x
term 3 does not have a variable base and an exponent
that is a whole number.
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
–
f (x) = 6x 2 + 2x 1 + x
SOLUTION
The function is not a polynomial function because the term
2x –1 has an exponent that is not a whole number.
Identifying Polynomial Functions
Polynomial function?
f (x) = 12 x 2 – 3x 4 – 7
f (x) = x 3 + 3x
f (x) = 6x2 + 2x– 1 + x
f (x) = – 0.5x +  x2 –
2
Polynomial Functions can be
classified by degree
Polynomial Functions can be classified by degree
and by the number of terms
f ( x)  3
CONSTANT,
MONOMIAL
f ( x)  3x  8
LINEAR,
BINOMIAL
f ( x)  3 x  8 x  1
QUADRATIC,
TRINOMIAL
f ( x)  2 x  4 x  3x  12
CUBIC,
POLYNOMIAL
2
3
2
f ( x)  2 x  4 x  3x  12
3
Given f(x) find f(-3).
-69
2
End Behavior Task
Let’s Summarize
GRAPHING POLYNOMIAL FUNCTIONS
CONCEPT
SUMMARY
END BEHAVIOR FOR POLYNOMIAL FUNCTIONS
x
+
+
f (x)
+
f (x)
–
f (x)
+
even
f (x)
–
f (x)
–
odd
f (x)
+
f (x)
–
an
n
x
>0
even
f (x)
>0
odd
<0
<0
–
Ex.
Determine the left and right behavior of the graph
of each polynomial function.
f(x) = x4 + 2x2 – 3x
f(x) = -x5 +3x4 – x
f(x) = 2x3 – 3x2 + 5
as x  _____ f ( x)  _____
as x  _____ f ( x)  _____
as x  _____ f ( x)  _____
as x  _____ f ( x)  _____
as x  _____ f ( x)  _____
as x  _____ f ( x)  _____
Tell me what you know about the equation…
Odd exponent
Positive leading coefficient
Tell me what you know about the equation…
Even exponent
Positive leading coefficient
Tell me what you know about the equation…
Odd exponent
Positive leading coefficient
Tell me what you know about the equation…
Even exponent
Negative leading coefficient