6.4 Polynomial Functions

• Polynomial in one variable : A polynomial with only one variable

• Leading coefficient: the coefficient of the term with the highest degree in a polynomial in one variable

• Polynomial Function: A polynomial equation where the y is replaced by f(x)

State the degree and leading cofficient of each polynomial, if it is not a polynomial in

one variable explain why.1. 7x4 + 5x2 + x – 9

2. 8x2 + 3xy – 2y2

3. 7x6 – 4x3 + x-1

4. ½ x2 + 2x3 – x5

Evaluating Functions

• Evaluate f(x) = 3x2 – 3x +1 when x = 3

• Find f(b2) if f(x) = 2x2 + 3x – 1

• Find 2g(c+2) + 3g(2c) if g(x) = x2 - 4

End Behavior

• Describes the behavior of the graph f(x) as x approaches positive infinity or negative infinity.

• Symbol for infinity

End behavior Practice

f(x) - as x +

f(x) - as x -

End Behavior Practice

f(x) + as x +

f(x) - as x -

End Behavior Practice

f(x) + as x +

f(x) + as x -

The Rules in General

To determine if a function is even or odd

• Even functions: arrows go the same direction• Odd functions: arrows go opposite directions

To determine if the leading coefficient is positive or negative

• If the graph goes down to the right the leading coefficient is negative

• If the graph goes up to the right then the leading coeffiecient is positive

The number of zeros • zeros are the same as roots: where the graph crosses the

x-axis– The number of zeros of a function can be equal to the exponent

or can be less than that by a multiple of 2.• Example a quintic function, exponent 5, can have 5, 3 or 1 zeros

• To find the zeros you factor the polynomial

Critical Points• points where the graph changes direction.

– These points give us maximum and minimum values• Relative Max/Min

Put it all together• For the graph given

– Describe the end behavior

– Determine whether it is an even or an odd degree

– Determine if the leading coefficient is positive or negative

– State the number of zeros

Cont…• For the graph given

– Describe the end behavior– Determine whether it is an

even or an odd degree– Determine if the leading

coefficient is positive or negative

– State the number of zeros

• For the graph given – Describe the end

behavior– Determine whether it

is an even or an odd degree

– Determine if the leading coefficient is positive or negative

– State the number of zeros

• For the graph given – Describe the end

behavior– Determine whether it is

an even or an odd degree

– Determine if the leading coefficient is positive or negative

– State the number of zeros