Rational functions asymptotes and excluded values

Learning Objectives Identify all excluded values of a rational function by finding the zeros of the denominator. Distinguish between removable discontinuities (holes) and non-removable discontinuities (vertical asymptotes). Determine the equations of all vertical asymptotes of a rational function. Determine the equation of the horizontal or slant (oblique) asymptote by comparing the degrees of the numerator and denominator polynomials. Use limit notation to describe the behavior of a rational function as it approaches its asymptotes. Synthesize information about asymptotes, holes, and intercepts to sketch an accurate graph of a rational function. Ever wondered what happens when a function's graph shoots off towards infinity? 🚀 We're about to explore the invisible...

TermDefinitionExample Rational FunctionA function that can be written as the ratio of two polynomial functions, f(x) = P(x) / Q(x), where Q(x) is not the zero polynomial.f(x) = (2x^2 + 1) / (x - 3) Excluded ValueAny value of x for which the denominator of a rational function equals zero, making the function undefined at that point.For f(x) = (x + 5) / (x - 2), the excluded value is x = 2. Vertical AsymptoteA vertical line, x = c, that the graph of a function approaches as the y-values approach positive or negative infinity. It occurs at an excluded value that does not cancel out.For f(x) = 1 / (x + 4), the vertical asymptote is the line x = -4. Hole (Removable Discontinuity)A single point at x = c where a function is undefined, but can be 'filled in'. It occurs when a common fac...

Finding Vertical Asymptotes and Holes Given f(x) = P(x) / Q(x), first factor P(x) and Q(x). If a factor (x - c) cancels, there is a hole at x = c. For any remaining factor (x - a) in Q(x), there is a vertical asymptote at x = a. This is the first step in analyzing any rational function. Factoring and simplifying reveals whether an excluded value corresponds to a hole or a vertical asymptote. Finding Horizontal Asymptotes Let f(x) = (a_n x^n + ...) / (b_m x^m + ...). Let n be the degree of the numerator and m be the degree of the denominator. Case 1: If n < m, the HA is y = 0. Case 2: If n = m, the HA is y = a_n / b_m. Case 3: If n > m, there is no horizontal asymptote. Use this rule to determine the end behavior of the function. Compare the highest powers of x in the n...