Download Algebra 2: Section 6.2

Algebra 2: Section 6.2
Evaluating and Graphing Polynomial
Functions
(Day 1)
1
Polynomial Function
• A function is a polynomial function if…
– Exponents are all whole numbers
– Coefficients are all real numbers
• Standard Form of Polynomial Function
– All terms are written in descending order of
exponents from left to right
f ( x)  an x  an 1 x
n
n 1

 a1 x  a0
2
Parts of Polynomial
Function
f ( x)  an x  an 1 x
n
n 1

an
• Leading coefficient
a0
• Constant term
 a1 x  a0
– Coefficient on highest power of x
– Term that has no variable (no x)
• Degree of the polynomial
– Exponent of the highest power of x
3
Classifying Polynomial
Functions
• Classify based on highest power of x
• Power of x (degree)
– Constant:
Degree = 0:
f(x) =
5
– Linear:
Degree = 1:
f(x) =
2x  3
– Quadratic: Degree = 2:
f(x) = 4 x
2
 2x  5
– Cubic:
Degree = 3:
f(x) = 4 x 3
 3x  1
– Quartic:
Degree = 4:
f(x) = 6 x 4  4 x 2  2 x  5
2
4
Examples
• 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.
1. f ( x)  2 x  x
2
2
No, because negative exponent.
5
Examples
2. g ( x)  0.8x  x  5
3
4
Yes
4
3
f ( x)  x  0.8 x  5
Degree: 4
Type: Quartic
Leading Coefficient: 1
6
3) f ( x)  x  3
3
x
No, because it has an "x as the exponent.
7
Evaluating a Polynomial
• Direct substitution – putting in the value
of x in place of x and solving
• Synthetic substitution – similar to
synthetic division but the answer is just
the last number of the problem.
8
Direct Substitution
3. f ( x)  3x  x  5x  10 when x  2
5
4
f (2)  3(2)  (2)  5(2)  10
5
4
f (2)  3(32)  (16)  5(2)  10
f (2)  96  16  10  10
f (2)   92
9
Synthetic Substitution
(Synthetic Division)
• Gives another way to evaluate a function
• Also used to divide polynomials
– This will be discussed in later sections
• The last column is the value of the
function
10
Examples
• Use synthetic division to evaluate.
3. f ( x)  3x  x  5x  10 when x  2
5
4
11
3. f ( x)  3x  x  5x  10 when x  2
5
Number you are
dividing by goes in
front
2 3
4
Coefficients of x written in order
1
Missing power of
x, zero coefficient!
0
0 5
10
6 14 28 56 102
3st 7 14 28 51 92
Drop 1
number
down
**Multiply Across…..
Add Down
ANSWER!
12
4. f (4)  5x  x  4 x  1
3
4 5
5
1
2
4
1
20
84 320
21
80 321
13
Homework
• P.333
16 - 26 evens
28 & 30
38 - 46 evens
14