6.2 Evaluating and Graphing Polynomial Functions

1
6.2 Evaluating and Graphing Polynomial Functions

• Goal
• Evaluate and graph polynomial functions.

2
Polynomial Function

• Function of the form
• an is the leading coefficient
• a0 is the constant term
• n is the degree
• Polynomial is in standard form if its terms are
  written in descending order of exponents from
  left to right.

3
A polynomial function is a function of the form
f (x) an x n an 1 x n 1 a 1 x
a 0
Where an ? 0 and the exponents are all whole
numbers.
For this polynomial function, an is the
leading coefficient, a 0 is the constant term,
and n is the degree.
A polynomial function is in standard form if
its terms are written in descending order of
exponents from left to right.
4
Common Types of Polynomials
Degree Type Standard Form
0 Constant f(x) a0
1 Linear f(x) a1x a0
2 Quadratic f(x) a2x2 a1x a0
3 Cubic f(x) a3x3 a2x2 a1x a0
4 Quartic f(x) a4x4 a3x3 a2x2 a1x a0
5
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.
6
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.
7
Evaluating a Polynomial Function

• One way to evaluate is to use direct substitution.

8
One way to evaluate polynomial functions is to
usedirect substitution. Another way to evaluate
a polynomialis to use synthetic substitution.
9
SOLUTION
2 x 4 0 x 3 (8 x 2) 5 x (7)
Polynomial in standard form
2 0 8 5 7




3
Coefficients
6
18
30
105
35
10
98
2
6
The value of f (3) is the last number you
write, In the bottom right-hand corner.
10
Evaluate using Direct Substitution and Synthetic
Substitution
Direct Substitution Synthetic Substitution
11
Evaluate the Polynomial FunctionUsing Synthetic
Substitution
12
Evaluating a Polynomial Function in Real Life

• The time t (in seconds) it takes a camera battery
  to recharge after flashing n times can be modeled
  by
• Find the recharge time after 100 flashes.

13
Homework

• Pg. 333-334 15 - 45 every 3rd Problem

14
6.2 Continued Graphing Polynomial Functions

• Will use end behavior to analyze the graphs of
  polynomial functions.

15
End Behavior

• Behavior of the graph as x approaches positive
  infinity (8) or negative infinity (-8)
• The expression x?8 as x approaches positive
  infinity
• The expression x?-8 as x approaches negative
  infinity

16
End Behavior of Graphs of Linear Equations
f(x) x
f(x) -x
f(x)?-8 as x?8 f(x)?8 as x?-8
f(x)?8 as x?8 f(x)?-8 as x?-8
17
End Behavior of Graphs of Quadratic Equations
f(x) x²
f(x) -x²
f(x)?-8 as x?8 f(x)?-8 as x?-8
f(x)?8 as x?8 f(x)?8 as x?-8
18
Investigating Graphs of Polynomial Functions

• Use a Graphing Calculator to grph each function
  then analyze the functions end behavior by
  filling in this statement f(x)?__8 as x?8 and
  f(x)?__8 as x?-8
• a. f(x) x³ c. f(x) x4 e. f(x) x5 g. f(x)
  x6
• b. f(x) -x³ d. f(x) -x4 f. f(x) -x5 h.
  f(x) -x6

19
Investigating Graphs of Polynomial Functions

• How does the sign of the leading coefficient
  affect the behavior of the polynomial function
  graph as x?8?
• How is the behavior of a polynomial functions
  graph as x?8 related to its behavior as x?-8
  when the functions degree is odd? When it is
  even?

20
End Behavior for Polynomial Functions

• For the graph of
• If angt0 and n even, then f(x)?8 as x?8 and
  f(x)?8 as x?-8
• If angt0 and n odd, then f(x)?8 as x?8 and
  f(x)?-8 as x?-8
• If anlt0 and n even, then f(x)?-8 as x?8 and
  f(x)?-8 as x?-8
• If anlt0 and n odd, then f(x)?-8 as x?8 and
  f(x)?8 as x?-8

21
Graphing Polynomial Functions

• f(x) x³ x² 4x 1

x -3 -2 -1 0 1 2 3
f(x)
22
Graphing Polynomial Functions

• f(x) -x4 2x³ 2x² 4x

x -3 -2 -1 0 1 2 3
f(x)
23
Homework

• Pg. 334 49-52, 53-71 every 3rd problem