{"id":1671,"date":"2021-05-26T12:06:39","date_gmt":"2021-05-26T06:36:39","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=1671"},"modified":"2021-10-30T07:40:42","modified_gmt":"2021-10-30T07:40:42","slug":"what-are-the-asymptotes","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-are-the-asymptotes\/","title":{"rendered":"What are the Asymptotes?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

A line that a curve approaches is known as asymptote. Any graph (curve) approaches to it but never touches it.<\/p>\n\n\n\n

There are total three types of asymptotes-<\/p>\n\n\n\n

1.     Vertical asymptote<\/p>\n\n\n\n

2.     Horizontal asymptote<\/p>\n\n\n\n

3.     Oblique\/slant asymptote<\/p>\n\n\n\n

Asymptotes<\/strong><\/h2>\n\n\n\n

An asymptote of a curve of function y = f(x) is a line which does not intersect on the graph.<\/p>\n\n\n\n

Types of asymptotes-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

1. Vertical asymptote-<\/strong><\/p>\n\n\n\n

A line x = a which is straight is a vertical asymptote of the graph of the function y = f(x) if atleast one of the following condition does it follow-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

2. Horizontal asymptote-<\/strong><\/p>\n\n\n\n

A line y = L is called horizontal asymptote to the curve of the function f(x)-<\/p>\n\n\n\n

If f(x)\u2192 L as x\u2192\u221e or as x\u2192-\u221e<\/p>\n\n\n\n

Example: Find the horizontal asymptote of the function-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Sol. In order to find the horizontal asymptote-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Hence the horizontal asymptote is y = 2.<\/p>\n\n\n\n

3. Oblique asymptote or slant asymptote-<\/strong><\/p>\n\n\n\n

A straight line y = mx + c where m \u2260 0 will be an oblique\/slant asymptote to the graph of the function \u2018f\u2019 if-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Note-<\/strong> the value of m can be find as-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

And the value of c can be find as-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Example: Find the slant asymptote of the function f(x) = x + <\/strong>x .<\/strong><\/p>\n\n\n\n

Sol. Find the value of m-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Hence the y = x + c,<\/p>\n\n\n\n

Now find c<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Here c must not be infinite.<\/p>\n\n\n\n

So we can say that f does not have a slant asymptote at \u221e.<\/p>\n\n\n\n

Procedure to find the asymptotes parallel to axes of a polynomial function-<\/strong><\/p>\n\n\n\n

Theorem-<\/strong><\/p>\n\n\n\n

Suppose f(x , y) is a polynomial in x and y.<\/p>\n\n\n\n

A straight line y = c is an asymptote of a curve f(x , y) = 0 if an only if y \u2013 c is a factor of the co-efficient of the highest power of x in f(x , y).<\/p>\n\n\n\n

Example: Determine the asymptotes parallel to axes of the curve:<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Sol. The given function can be written as-<\/p>\n\n\n\n

\"\"\/<\/figure><\/div>\n\n\n\n

Asymptote parallel to x-axis-<\/strong><\/p>\n\n\n\n

Equating the coefficient of the highest power of x to zero, we get y\u00b2 = 0 which means y = 0. This is an asymptote.<\/p>\n\n\n\n

Asymptote parallel to y-axis-<\/strong><\/p>\n\n\n\n

Equating the coefficient of the highest power of x to zero, we get x\u00b2 – a\u00b2 = 0<\/p>\n\n\n\n

Which gives x = \u00b1a , that means x = +a , x = -a     <\/p>\n\n\n\n

So that these are the asymptotes.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n