Section 1.6 Powers, Polynomials, and Rational Functions

1. Section 1.6Powers, Polynomials, and Rational Functions

2. Often in this class we will deal with functions of the form • Functions of this form are called power functions • Notice the variable is being raised to an exponent • Contrast this with an exponential function where the variable is in the exponent

3. Which of the following are power functions and identify the k and the n (recall )

4. Power functions can be odd, even or neither • How can we decide? • What about the following? • What about the end behavior of a power function versus an exponential • Which grows faster?

5. What happens if we add or subtract power functions? • A polynomial is a sum (or difference) of power functions whose exponents are nonnegative integers • What determines the degree of a polynomial? • For example • What is the leading term in this polynomial?

6. We have the general form of a polynomial which can be written as • Where n is a positive integer called the degree of p • Each power function is called a term • The constants an , an-1,… a0,are called coefficients • The term a0 is called the constant term • The term anxn is called the leading term

7. End Behavior The shape of the graph of a polynomial function depends on the degree. Degree EVENDegree ODD an>0 an<0 an>0 an<0

8. What are the zeros (or roots) of a polynomial? • Where the graph hits the x-axis • The input(s) that make the polynomial equal to 0 • How can we find zeros of a polynomial? • For example, what are the zeros of • Notice this polynomial is in its factored form • It is written as a product of its linear factors • A polynomial of degree n can have at most n real zeros

9. Behavior of Polynomials x x What is the significance of this point? What is the significance of this point? What behavior do you notice at the zeros of these functions?

10. When a polynomial, p, has a repeated linear factor, then it has a multiple root • If the factor (x - k) is repeated an even number of times, the graph does not cross the x-axis at x = k. It ‘bounces’ off. The higher the (even) exponent, the flatter the graph appears around x = k. • If the factor (x - k) is repeated an odd number of times, the graph does cross the x-axis at x = k. It appears to flatten out. The higher the (odd) exponent, the flatter it appears around x = k.

11. If r can be written as the ratio of polynomial functions p(x) and q(x), then r is called a rational function • The long-run behavior is determined by the leading terms of both p and q • These functions often have horizontal asymptotes which define their long run behavior

12. We have three cases • The degree of p < the degree of q • The horizontal asymptote is the line y = 0 • The degree of p > the degree of q • There is no horizontal asymptote • The degree of p = the degree of q • The horizontal asymptote is the ratio of the coefficients of the leading terms of p and q

13. Let’s consider the following functions • How do we find their x-intercepts? • What are they? • What happens if the denominators equal 0? • What are their horizontal asymptotes?