Post on 27-Mar-2015
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
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 ?
A polynomial function is a function of the form:
on
nn
n axaxaxaxf 1
11
All of these coefficients are real numbers
n must be a positive integer
Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 …
The degree of the polynomial is the largest power on any x term in the polynomial.
Polynomial Functions
• Exponents must be non-negative integer exponents
• Can not have variables in the denominator
• Can not have radicals – Example: square root or cube root– These are actually fractional exponents
x 0
2
1
xx
Not a polynomial because of the square root since the power is NOT an integer
xxxf 42
Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.
A polynomial of degree 4.
2xg
12 xxh
23x
xxF
A polynomial of degree 0.
We can write in an x0 since this = 1.
Not a polynomial because of the x in the denominator since the power is negative 11 x
x
Graphs of polynomials are smooth and continuous.
No sharp corners or cusps No gaps or holes, can be drawn without lifting pencil from paper
This IS the graph of a polynomial
This IS NOT the graph of a polynomial
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
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = 3
Constant Function
Degree = 0
Max. Zeros: 0
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x + 2
LinearFunction
Degree = 1
Max. Zeros: 1
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QuadraticFunction
Degree = 2
Max. Zeros: 2
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CubicFunction
Degree = 3
Max. Zeros: 3
POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
QuarticFunction
Degree = 4
Max. Zeros: 4
LEFT RIGHTand
HAND BEHAVIOUR OF A GRAPH
The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behaviour of a graph.
Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term.
left hand behaviour: rises
right hand behaviour: rises
Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative.
left hand behaviour: falls right hand
behaviour: falls
turning points in the middle
Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive.
left hand behaviour: falls
right hand behaviour: rises
turning Points in the middle
Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative.
left hand behaviour: rises
right hand behaviour: falls
turning points in the middle
A polynomial of degree n can have at most n-1 turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn).
doesn’t mean it has that many turning points but that’s the most it can have
3019153 234 xxxxxf
Let’s determine left and right hand behaviour for the graph of the function:
degree is 4 which is even and the coefficient is positive so the graph will look like x2 looks off to the left and off to the right.
The graph can have at most 3 turning points
How do we determine
what it looks like near the
middle?
Characteristics• Maximum number of turns in 1 less than
the degree
• Degree– Odd with positive leading coefficient
• Starts down and comes up
– Even with positive leading coefficient• Starts up and comes down
• Negative leading coefficient changes direction of starting position
x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x.
3019153 234 xxxxxf
30300190150300 234 f
(0,30)
To find the x intercept we put 0 in for y.
51320 xxxx
Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that’s okay)