7.6 Rational Functions

A rational function is a quotient of 2 polynomials A rational function can have places where the graph is not defined There are 3 types:  where the denominator is 0 point(a hole in a graph) get when you cancel out a factor jump(the graph jumps) infinite (it has a vertical asymptote) A rational function can also have a horizontal asymptote

Let’s explore one of these with our graphing calculators & talk about some notation. Looks like a vertical asymptote at x = 2 We write (x approaches 2 from the right)(x approaches 2 from the left) And a horizontal asymptote at y = 0 and *Use trace and see what the graph is going towards We write and (graph on TV)

Asymptote Rules (MEMORIZE): Let be a rational function where f (x) is a polynomial of degree n and g(x) is a polynomial of degree m. – If g(a) = 0 and f (a) ≠ 0, then x = a is a vertical asymptote – Horizontal asymptotes have 3 possible conditions: If n < m, then y = 0 is horizontal asymptote If n > m, then no horizontal asymptote If n = m, then y = c is horizontal asymptote, where c is the quotient of leading coefficients of f and g *You need to know how to do it by hand, but it won’t hurt to confirm your answer with your graphing calculator

Ex 1) Graph & determine points of discontinuity x ≠ –4 *line with a hole at x = –4

Ex 2) Determine the intercepts and asymptotes and graph x ≠ 5vert. asympt. degree same  horiz. asympt. at intercepts  y-int (0, ___) x-int (___, 0)

Ex 3) Graph & determine vertical & horizontal asymptotes x ≠ 3 & x ≠ –2 vert. asympt. at x = 3 & x = –2 degree same  horiz. asympt. at x-int at x = 2 & x = –1 y-int at y = ⅓

We can also have slant asymptotes! (when degree of f is exactly 1 more than degree of g) Asymptote is quotient you get when you divide Ex 4) Graph & determine asymptotes: x ≠ –3 Asympt: y = 4x – 5 deg of numerator is 1 more than degree of denom

Homework #706 Pg 368 #1, 5, 9, 13, 16, 22, 23, 28, 30, 36, 40, 42, 44, 51, 54–57