how do you find asymtotes equations of a graph
Answers
Answer:
A function of the form where t(x) and n(x) are polynomials is called a rational function.
The graphs of rational functions can be recognised by the fact that they often break into two or more parts. These parts go out of the coordinate system along an imaginary straight line called an asymptote.
Let's look at the function
This graph follows a horizontal line ( red in the diagram) as it moves out of the system to the left or right. This is a horizontal asymptote with the equation y = 1. As x gets near to the values 1 and –1 the graph follows vertical lines ( blue). These vertical asymptotes occur when the denominator of the function, n(x), is zero ( not the numerator).
To find the equations of the vertical asymptotes we have to solve the equation:
x2 – 1 = 0
x2 = 1
x = 1 or x = –1
Near to the values x = 1 and x = –1 the graph goes almost vertically up or down and the function tends to either +∞ or –∞.
We get a horizontal asymptote because the numerator and the denominator, t(x) = x2 and n(x) = x2 – 1 are almost equal as x gets bigger and bigger.
If, for example, x = 100 then x2 = 10000 and x2 – 1 = 9999 , so that when we divide one by the othere we get almost 1. The bigger the value of x the nearer we get to 1.
Vertical asymptotes can be found by solving the equation n(x) = 0 where n(x) is the denominator of the function ( note: this only applies if the numerator t(x) is not zero for the same x value).
Horizontal asymptotes can be found by finding the limit
Example 1
Find the asymptotes for the function .
To find the vertical asymptote we solve the equation
x – 1 = 0
x = 1
The graph has a vertical asymptote with the equation x = 1.
To find the horizontal asymptote we calculate .
The numerator always takes the value 1 so the bigger x gets the smaller the fraction becomes. For example if x = 1000 then f(x) = 001. As x gets bigger f(x) gets nearer and nearer to zero.
This tells us that y = 0 ( which is the x-axis ) is a horizontal asymptote.
Finally draw the graph in your calculator to confirm what you have found.
The above example suggests the following simple rule:
A rational function in which the degree of the denominator is higher than the degree of the numerator has the x axis as a horizontal asymptote.