{"id":1758,"date":"2012-09-12T23:39:15","date_gmt":"2012-09-12T23:39:15","guid":{"rendered":"http:\/\/testpreparations.com\/help\/?p=1758"},"modified":"2014-12-02T08:32:03","modified_gmt":"2014-12-02T08:32:03","slug":"asymptotes-of-rational-functions","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/asymptotes-of-rational-functions\/","title":{"rendered":"Asymptotes of Rational Functions"},"content":{"rendered":"

An asymptote of a function is a line where the length between the function and the line approach but do not reach zero as the function continues to infinity. There are three types of asymptotes: horizontal, vertical and oblique.<\/p>\n

Vertical Asymptotes<\/h4>\n

<\/strong>Vertical asymptotes occur whenever the denominator of a function is equal to zero as we can never divide by the number zero. In order to find the vertical asymptotes, we simply set the denominator equal to zero and solve for x.<\/p>\n

Example:<\/span><\/p>\n

Find the vertical asymptotes of<\/strong><\/p>\n

\"\"<\/a><\/p>\n

So we set x2<\/sup> – 2x + 3 = 0 and solve.<\/p>\n

x2<\/sup> – 2x + 3 = 0<\/p>\n

(x + 3)(x – 1) = 0<\/p>\n

x = -3, x = 1<\/p>\n

Therefore our vertical asymptotes are x = -3 and x = 1.<\/p>\n

Horizontal Asymptotes<\/h4>\n

<\/strong>Horizontal asymptotes occurs when the degree of the denominator is greater than or equal to the degree of the numerator. If the degree of the denominator is equal than the degree of the numerator, then there is a horizontal asymptote. The ratio of the leading coefficients is the equation of the horizontal asymptote. If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0.<\/p>\n

Examples:<\/span><\/p>\n

Find the horizontal asymptotes of<\/strong><\/p>\n

\"\"<\/a><\/p>\n

We can see the degree of the denominator is greater than the degree of the numerator, and thus the equation of the horizontal asymptote is simply y = 0.<\/p>\n

 <\/p>\n

\"\"<\/a><\/p>\n

We can see the degree of the denominator is equal to the degree of the numerator, and thus we take the ratio of the coefficients. The ratio is 3\/4 and thus our horizo<\/strong>ntal asymptote is at y = 3\/4<\/strong><\/p>\n

Oblique<\/strong><\/span> Asym<\/strong><\/strong>ptotes<\/strong><\/span><\/p>\n

Oblique asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. Then the asymptote occurs at the division of the numerator by the denominator and taking the polynomial section ignoring the remainder.<\/p>\n

Example:<\/span><\/p>\n

Find the oblique asymptote of<\/strong><\/p>\n

<\/strong>\"\"<\/a><\/p>\n

We can see the degree of the numerator is in fact one higher than the degree of the denominator, thus this function has an oblique asymptote. The asymptote is then found at y = x + 1 because we divide each term by x and take the polynomial section ignoring the remainder of 1\/x.<\/p>\n

 <\/p>\n

This article was written for you by Jeremie<\/strong>, one of the tutors with Test Prep Academy.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

An asymptote of a function is a line where the length between the function and the line approach but do not reach zero as the function continues to infinity. There are three types of asymptotes: horizontal, vertical and oblique. Vertical Asymptotes Vertical asymptotes occur whenever the denominator of a function is equal to zero as […]<\/p>\n","protected":false},"author":6,"featured_media":2108,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2841,3015,3021,2851],"tags":[2666,3109,3228,3261,3343],"class_list":["post-1758","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-act","category-math-act","category-mathematics-sat","category-sat","tag-asymptotes","tag-horizontal-asymptotes","tag-oblique-asymptotes","tag-rational-functions","tag-vertical-asymptotes"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/1758","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=1758"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/1758\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media\/2108"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=1758"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=1758"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=1758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}