{"id":8543,"date":"2017-01-13T09:10:29","date_gmt":"2017-01-13T17:10:29","guid":{"rendered":"https:\/\/magoosh.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":"https:\/\/magoosh.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? 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