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7.6 Rational Functions

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. get when you cancel out a factor. point. (a hole in a graph). jump. (the graph jumps).

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7.6 Rational Functions

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  1. 7.6 Rational Functions

  2. A rationalfunction 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 get when you cancel out a factor point (a hole in a graph) jump (the graph jumps) infinite (it has a vertical asymptote) A rational function can also have a horizontal asymptote

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

  4. 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 verticalasymptote – Horizontalasymptotes 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

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

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

  7. 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

  8. 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 deg of numerator is 1 more than degree of denom Asympt: y = 4x – 5

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

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