5.2
Evaluate and Graph Polynomial Functions
Polynomial
A monomial or a sum
of monomials
Polynomial function
A function of the
form f(x) = anxn
+ an-1 xn-1 + ¼ + a1x + a0, where an
¹ 0, the
exponents are all whole numbers, and the coefficients are all real numbers
Synthetic substitution
An alternate method
to evaluate a polynomial function using fewer operations than direct
substitution
End behavior
The behavior of a
polynomial function's graph as ^approaches positive infinity or negative
infinity
Example
1
Identify polynomial
functions
Decide whether the
function is a polynomial function. If so, write it in standard form and state
its degree, type, and leading coefficient.
a. f(x)
= 3x3 + 4x2..5 - 6x2 b. f(x) = x2
+ 3.7x + 9x4
Solution
a.
The
function _is not_ a polynomial function because the term _4 x 2.5_ has
an exponent that is _not a whole number_.
b.
The
function _is_ a polynomial function written as _f(x) = 9x4 + x2 + 3.7x_ in its standard form. It has degree _4_
(_quartic_) and a leading coefficient of _9_.
State the degree,
type, and leading coefficient of the function.
1. f(x) = -2x3 + 2x2
- 3x4
+ 5
degree: 4, type: quartic, leading coefficient: -3
Example 2
Evaluate by synthetic
substitution
Use synthetic
substitution to evaluate f(x) = 2x4 + 3x3
- 6x2 + 3 when x = 2.
Write the coefficients of f(x)
in order of _descending_ exponents. Write the value of x to
the left. Bring down the leading coefficient. Multiply the leading coefficient
by _2_ and write the product under the second
coefficient. _Add_ . Multiply the previous sum by _2_ and
write the product under the second coefficient. Add. Repeat for all of the
remaining coefficients.
_2_ coefficients
f(2) = _35_
END BEHAVIOR OF POLYNOMIAL
FUNCTIONS
For the graph of
f(x) = anxn
+ an-1 xn-1 + ¼ + a1x + a0:
• If an
> 0 and n is odd, then f(x) ® _-¥_ as
x ® -¥ and f(x) ® _+¥_ as x ® + ¥.
• If an
< 0 and n is odd, then f(x) ® _+¥_ as
x ® -¥ and f(x) ® _-¥_ as x ® + ¥.
• If an
> 0 and n is even, then f(x) ® _+¥_ as
x ® -¥ and f(x) ® _+¥_ as x ® + ¥.
• If an
< 0 and n is even, then f(x) ® _-¥_ as
x ® -¥ and f(x) ® _-¥_ as x ® + ¥.
Example
3
Graph polynomial
functions
Graph f(x)
= -x3 + 2x2 + 2x - 1.
Solution
Make a table of
values and plot the corresponding points. Connect the points with a smooth
curve and check the end behavior.
|
x
|
-3
|
-2
|
-1
|
0
|
1
|
2
|
3
|
|
f(x)
|
_38_
|
_11_
|
_0_
|
_-1_
|
_2_
|
_3_
|
_-4_
|
The degree is _odd_
and the leading coefficient is _negative_, so f(x) ® _+¥_ as
x ® - ¥ and f(x) ® _- ¥_ as x ® +¥.
Complete the
following exercises using the function f(x) = -x4 + 3x3 + x2
- 4x - 1.
2.
Evaluate f(x) for x
= -2
using synthetic substitution.
f( -2) = -29
3.
Graph f(x).
