{"id":8543,"date":"2017-01-13t09:10:29","date_gmt":"2017-01-13t17:10:29","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8543"},"modified":"2017-06-19t13:59:25","modified_gmt":"2017-06-19t20:59:25","slug":"oblique-asymptotes","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/oblique-asymptotes\/","title":{"rendered":"how do you find the oblique asymptotes of a function?"},"content":{"rendered":"

in my experience, students often hit a roadblock when they see the word asymptote<\/em>. what is an asymptote anyway? how do you find them? is this going to be on the test??? (the answer to the last question is yes<\/em>. asymptotes definitely show up on the ap calculus exams<\/a>).<\/p>\n

of the three varieties of asymptote — horizontal<\/a>, vertical<\/a>, and oblique — perhaps the oblique asymptotes are the most mysterious. in this article we define oblique asymptotes and show how to find them.<\/p>\n

what is an oblique asymptote?<\/h2>\n

an oblique<\/strong> (or slant<\/strong>) asymptote<\/strong> is a slanted line that the function approaches as x<\/em> approaches \u221e (infinity<\/em>) or -\u221e (minus infinity<\/em>). let’s explore this definition a little more, shall we?<\/p>\n

it’s all about the line<\/h3>\n

since all non-vertical lines can be written in the form y<\/em> = mx<\/em> + b<\/em> for some constants m<\/em> and b<\/em>, we say that a function f<\/em>(x<\/em>) has an oblique asymptote y<\/em> = mx<\/em> + b<\/em> if the values (the y<\/em>-coordinates) of f<\/em>(x<\/em>) get closer and closer to the values of mx<\/em> + b<\/em> as you trace the curve to the right (x<\/em> \u2192 \u221e) or to the left (x<\/em> \u2192 -\u221e), in other words, if there is a good approximation<\/strong>,<\/p>\n

f<\/em>(x<\/em>) ≈ mx<\/em> + b<\/em>,<\/p>\n

when x<\/em> gets extremely large in the positive or negative sense.<\/p>\n

still with me? i understand completely if you’re still a little lost, but let’s see if we can clear up some confusion using the graph shown below.<\/p>\n

\"oblique<\/p>\n

as you can see, the function (shown in blue) seems to get closer to the dashed line. therefore, the oblique asymptote for this function is y<\/em> = ½ x<\/em> – 1.<\/p>\n

finding oblique aymptotes<\/h2>\n

a function can have at most two oblique asymptotes, but only certain kinds of functions are expected to have an oblique asymptote at all. for instance, polynomials<\/em> of degree 2 or higher do not have asymptotes of any kind. (remember, the degree<\/strong> of a polynomial is the highest exponent on any term. for example, 10x<\/em>3<\/sup> – 3x<\/em>4<\/sup> + 3x<\/em> – 12 has degree 4.)<\/p>\n

as a quick application of this rule, you can say for sure without any work<\/em> that there are no oblique asymptotes for the quadratic function f<\/em>(x<\/em>) = x<\/em>2<\/sup> + 3x<\/em> – 10, because it’s a polynomial of degree 2.<\/p>\n

on the other hand, some kinds of rational functions<\/em> do have oblique asymptotes.<\/p>\n

rational functions<\/h3>\n

a rational function<\/strong> has the form of a fraction, f<\/em>(x<\/em>) = p<\/em>(x<\/em>) \/ q<\/em>(x<\/em>), in which both p<\/em>(x<\/em>) and q<\/em>(x<\/em>) are polynomials. if the degree of the numerator (top) is exactly one greater than the degree of the denominator (bottom), then f<\/em>(x<\/em>) will have an oblique asymptote. <\/p>\n

so there are no oblique asymptotes for the rational function, \"rational.<\/p>\n

but a rational function like \"rational does have one. knowing when there is a horizontal asymptote is just half the battle. now how do we find it? this next step involves polynomial division<\/em>.<\/p>\n

polynomial division to find oblique asymptotes<\/h3>\n

if you’ve made it this far, you probably have seen long division of polynomials, or synthetic division, but if you are rusty on the technique, then check out this video<\/a> or this article<\/a>.<\/p>\n

the idea is that when you do polynomial division on a rational function that has one higher degree on top than on the bottom, the result always has the form mx<\/em> + b<\/em> + remainder term<\/em>. then the oblique asymptote is the linear part, y<\/em> = mx<\/em> + b<\/em>. we don’t need to worry about the remainder term at all.<\/p>\n

example using polynomial division<\/h4>\n

let’s see how the technique can be used to find the oblique asymptote of \"rational.<\/p>\n

the long division is shown below. <\/p>\n

\"polynomial<\/p>\n

because the quotient is 2x<\/em> + 1, the rational function has an oblique asymptote:
\ny<\/em> = 2x<\/em> + 1.<\/p>\n

hyperbolas<\/h3>\n

another place where oblique asymptotes show up is in the graphs of hyperbolas<\/em>. remember, in the simplest case, a hyperbola<\/strong> is characterized by the standard equation,<\/p>\n

\"hyperbola<\/p>\n

the hyperbola graph corresponding to this equation has exactly two oblique asymptotes,<\/p>\n

\"equations<\/p>\n

the two asymptotes cross each other like a big x.<\/p>\n

example involving a hyperbola<\/h4>\n

let’s find the oblique asymptotes for the hyperbola with equation x<\/em>2<\/sup>\/9 – y<\/em>2<\/sup>\/4 = 1.<\/p>\n

\"hyperbola<\/p>\n

in the given equation, we have a<\/em>2<\/sup> = 9, so a<\/em> = 3, and b<\/em>2<\/sup> = 4, so b<\/em> = 2. this means that the two oblique asymptotes must be at y<\/em> = ±(b<\/em>\/a<\/em>)x<\/em> = ±(2\/3)x<\/em>.<\/p>\n

more general hyperbolas<\/h3>\n

it’s important to realize that hyperbolas come in more than one flavor. if the hyperbola has its terms switched, so that the “y<\/em>” term is positive and “x<\/em>” term is negative, then the asymptotes take a slightly different form. furthermore, if the center of the hyperbola is at a different point than the origin, (h<\/em>, k<\/em>), then that affects the asymptotes as well. below is a summary of the various possibilities.<\/p>\n

\"asymptotes_for_hyperbolas\"<\/p>\n

final thoughts<\/h2>\n

so when you see a question on the ap calculus ab exam asking about oblique asymptotes, don’t forget:<\/p>\n

    \n
  • if the function is rational, and if the degree on the top is one more than the degree on the bottom: use polynomial division.<\/li>\n
  • if the graph is a hyperbola with equation x<\/em>2<\/sup>\/a<\/em>2<\/sup> – y<\/em>2<\/sup>\/b<\/em>2<\/sup> = 1, then your asymptotes will be y<\/em> = ±(b<\/em>\/a<\/em>)x<\/em>. other kinds of hyperbolas also have standard formulas defining their asymptotes.\n<\/ul>\n

    keeping these techniques in mind, oblique asymptotes will start to seem much less mysterious on the ap exam!<\/p>\n","protected":false},"excerpt":{"rendered":"

    oblique asymptotes. what are they? how do you find them? and what good are they? find out about rational functions, hyperbolas and other special cases here!<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"class_list":["post-8543","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"\nhow do you find the oblique asymptotes of a function? - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"oblique asymptotes. what are they? how do you find them? and what good are they? find out about rational functions, hyperbolas and other special cases here!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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