Polynomial Function

1. POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial
or a sum of monomials.
A POLYNOMIAL IN ONE
VARIABLE is a polynomial that
contains only one variable.
Example: 5x2 + 3x - 7

2. POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is
the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient
of the term with the highest degree.
What is the degree and leading
coefficient of 3x5 – 3x + 2 ?

3. POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called
LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called
QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called
CUBIC POLYNOMIAL FUNCTIONS

4. POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(-2) if f(x) = 3x2 – 2x – 6
f(-2) = 3(-2)2 – 2(-2) – 6
f(-2) = 12 + 4 – 6
f(-2) = 10

5. POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find f(2a) if f(x) = 3x2 – 2x – 6
f(2a) = 3(2a)2 – 2(2a) – 6
f(2a) = 12a2 – 4a – 6

6. POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = 3
Constant
Function
Degree = 0
Max. Zeros: 0

7. POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1

8. POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2

9. POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3
Cubic
Function
Degree = 3

10. POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4

11. POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1
Quintic
Function
Degree = 5
Max. Zeros: 5

12. POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  +∞
f(x) = x2

13. POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Even
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  -∞
f(x) = -x2

14. POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  +∞
f(x) = x3

15. POLYNOMIAL FUNCTIONS
END BEHAVIOR
Degree: Odd
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  -∞
f(x) = -x3