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 Critical Points 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