Horizontal Asymptote Rules

Algebra 2 · Axiom Academy

LESSON Horizontal Asymptote Rules Find any rational function's horizontal asymptote by comparing degrees — and see why only the leading terms decide the end behavior. 1. What a Horizontal Asymptote Is A horizontal asymptote is a horizontal line the graph approaches as x runs off toward or . Take : push x higher and higher and the outputs crowd in on y = 2 , edging closer forever without ever landing on it. Its end behavior names the asymptote Write the function as and compare with . Three outcomes cover every rational function: The bottom grows faster, so the value is squeezed to zero : the asymptote is y = 0 . Top and bottom grow at the same rate, so the value settles at the ratio of the leading coefficients . The top outgrows the bottom without bound, so there is no horizontal asymptote (the graph keeps climbing). 3. Why Only the Leading Terms Matter As x grows, the highest-degree term dwarfs everything below it, so the lower terms stop mattering for end behavior. The whole function ends up behaving like its leading term over its leading term. Now it is just three quick reads: find the leading term on top and bottom, compare their degrees, and name the asymptote. No need to simplify the whole function. Numerator degree 1 , denominator degree 2 . The top degree is smaller, so use Case 1 : y = 0 . Both degrees are 2 . Equal degrees, so use Case 2 : . Numerator degree 3 , denominator degree 1 . The top degree is bigger, so Case 3 : no horizontal asymptote.

This is the written version of the interactive lesson above. See the full Algebra 2 course.