Bellwork 2. Find all zeros of the function, write the polynomial as a product of linear factors. 1. Find a polynomial function with integer coefficients that has the given zeros. 4, 3i

Last Nights Homework 53.-3/2, ±5i 55.-3, 5, ±2i 65. No, Setting h = 64 and solving the resulting equation yields imaginary roots,

2.6 Rational Functions and Asymptotes -How to find domains of rational functions? -How to find horizontal and vertical asymptotes of graphs of rational functions?

Rational Functions and Asymptotes A rational function can be written in the form The most basic rational function Where N(x) and D(x) are polynomials Domain: (-∞,∞) Horizontal Asymptote: x = 0 Vertical Asymptote: y = 0

The line x = a is a vertical asymptote of the graph of f if f(x)→∞ or f(x)→-∞ as x→a, either from the right or the left. The line y = b is a horizontal asymptote of the graph of f if f(x)→b as x→∞ or x→-∞.

Vertical Asymptotes Let f be the rational function Where N(x) and D(x) have no common factors. The graph of f has vertical asymptotes at the zeros of D(x).

Example 1: Find the Vertical Asymptotes. Hole at x = -2 No VA VA at x = 0, Hole at x = 3 VA at x = 5 Hole at x = 4

Horizontal Asymptotes The graph f has at most one horizontal asymptote determined by looking at the exponents of the numerator and the denominator. If n < m, then y = 0 is the H.A. If n = m, then y = a/b is the H.A. If n > m, then there is no H.A.

Example 2: Find the H.A. of the following functions. Bigger exponent in D(x). H.A. y = 0 Same exponent in N(x) and D(x). H.A. y = 2/3 Bigger exponent in N(x). No H.A.

Example 3: Find a functions vertical asymptotes, and horizontal asymptotes.

Tonight’s Homework Pg 195 #7-19. #40, 41