Rational functions: asymptotes and excluded values

Learning Objectives Define a rational function and identify its domain. Determine the excluded values of a rational function by finding the zeros of its denominator. Algebraically find the equations of vertical asymptotes. Algebraically find the equation of the horizontal asymptote by comparing the degrees of the numerator and denominator. Distinguish between a vertical asymptote and a hole (removable discontinuity). Describe the end behavior of a rational function using its horizontal asymptote. Ever tried to walk towards a wall by always covering half the remaining distance? You get closer and closer but never actually touch it! ♾️ Rational functions have similar boundaries called asymptotes. This tutorial explores rational functions, which are fractions made of polynomia...

TermDefinitionExample Rational FunctionA function that can be written as the ratio of two polynomial functions, P(x) and Q(x), in the form f(x) = P(x) / Q(x), where Q(x) cannot be the zero polynomial.f(x) = (3x + 2) / (x^2 - 5) is a rational function. Excluded ValueA value of 'x' for which a rational function is undefined. This occurs when the denominator, Q(x), is equal to zero.For f(x) = 1 / (x - 3), the excluded value is x = 3, because it would make the denominator zero. Vertical AsymptoteA vertical line, x = c, that the graph of a function approaches but never touches or crosses. It occurs at an excluded value that does not cancel out from the denominator.The function f(x) = 1 / (x - 3) has a vertical asymptote at x = 3. Horizontal AsymptoteA horizontal line, y = b, that the...

Finding Vertical Asymptotes and Holes For a simplified rational function f(x) = P(x) / Q(x), any value 'c' such that Q(c) = 0 corresponds to a vertical asymptote at x = c. If a factor (x-c) cancels from the numerator and denominator, there is a hole at x = c. First, factor the numerator and denominator completely. Cancel any common factors to identify holes. The zeros of the remaining factors in the denominator correspond to vertical asymptotes. Finding Horizontal Asymptotes Given f(x) = (ax^n + ...) / (bx^m + ...), where 'n' is the degree of the numerator and 'm' is the degree of the denominator: 1. If n < m, the HA is y = 0. 2. If n = m, the HA is y = a/b (ratio of leading coefficients). 3. If n > m, there is no horizontal asymptote....