6 4 Polynomial Functions Polynomial in one variable

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. 7 x 4 + 5 x 2 + x – 9 2. 8 x 2 + 3 xy – 2 y 2 3. 7 x 6 – 4 x 3 + x-1 4. ½ x 2 + 2 x 3 – x 5

Evaluating Functions • Evaluate f(x) = 3 x 2 – 3 x +1 when x = 3 • Find f(b 2) if f(x) = 2 x 2 + 3 x – 1 • Find 2 g(c+2) + 3 g(2 c) if g(x) = x 2 - 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