R ATIONAL F UNCTIONS AND A SYMPTOTES

W HAT IS A R ATIONAL F UNCTION ? It is a function that can be written in the form p(x)/q(x) where p and q are both polynomial functions, q(x) ≠ 0. Example: f(x) = 1 x What is the domain of f(x)? Answer: All reals except 0.

V ERTICAL A SYMPTOTES What happens to f(x) as x gets close to zero? From the right, f(x) increases without bound, so lim f(x) = ∞ x → 0 + From the left, f(x) decreases without bound, so lim f(x) = -∞ x → 0 - The line x = 0 is a vertical asymptote for this function.

H ORIZONTAL A SYMPTOTES The graph of f(x) = 1/x also has a horizontal asymptote. What is its equation? How would you know this without knowing what the graph looked like? Answer: y = 0. We could know this because when x is very large or very small y gets close to zero. Therefore lim f(x) = 0 and lim f(x) = 0 as x → ∞ x → -∞

F INDING A SYMPTOTES Now let’s decide from the equations what the horizontal asymptotes (why are there no vertical asymptotes?) are for these graphs. a)y = 2x/(3x 2 + 1) b)y = 2x 2 /(3x 2 + 1) c)y = 2x 3 /(3x 2 + 1) Check your answers by graphing each function.

Find all of the asymptotes for these graphs. y = 3x/(x 2 – 4) y = 2x/(x – 3) y = (x 2 – x – 6)/(x 2 – x – 2) Check your answers by graphing each function.

S UMMARY For a rational function f(x) = p(x)/q(x) 1.The graph has vertical asymptote(s) at the zeros of q(x). 2.The graph has at most one horizontal asymptote. a) If the degree of p is higher than the degree of q, there is not one. b) If the degree of p is equal to the degree of q, the asymptote is y = the ratio of the leading coefficients. c) If the degree of p is less than the degree of q, then the asymptote is y = 0..