Graphing Rational Functions and Finding Asymptotes Within the Function

by Gabriella Dalton, Algebra and Calculus CAPS Tutor

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A rational function is a fraction of polynomials. That is, if g(x) and p(x) are polynomials, then

g(x)/p(x)

is a rational function. The numerator is g(x) and the denominator is p(x).

Examples:

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All of these examples are rational functions and for each example there are different methods to graphing each function. All polynomials are unique to their functions and when a polynomial is divided by a polynomial, in other words a rational function, there are three specific methods to keep in mind. Below I will thoroughly go over each method.

Step-by-Step Methods to Finding an Asymptote:

In general… If you have a function that looks like the following:

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then:

1.      If n m, then the x-axis is the horizontal asymptote. (y=0)

2.      If n = m, then the horizontal asymptote is equal to the ratio of the leading coefficients.

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In this case, the horizontal asymptote would be: f(x)=3/11.

1.      If n m, then there is no horizontal asymptote and therefore is a slant (oblique) asymptote.

The next step to graphing a rational function is finding the domain of the function:

Finding the Domain of a Function:

In my opinion, finding the domain and range of functions can be the most daunting task when trying to graph a function. However, with a few steps you will be able to find the domain and range of any function with confidence and ease.

Example:

image

Step One: Evaluating the Problem Before Any Computations

Think and ask yourself if any of the “x-guys” are going to cause a problem. In other words, look at your function and ask yourself if any of the x’s are going to make your function undefined. In this case, this example is a function that should make you smile immediately. The reason being is when dealing with a polynomial your domain will always be the set of all real numbers and there will never, ever be anything to disrupt the continuity of your function. Of course unless you alter the polynomial you are working with and make it resemble something below:

image

if this is the case, then… Move on to step two!

Step Two: Finding the Domain

What happens if x is equal to three?

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So, in this case, x=3 is a bad guy and is going to make our function undefined. Is there any other number that will also make this function behave this way?

No! Great answer!! J So, here x 3 or else the function will be undefined. So, the domain will be all real numbers except for three. If we wish to express this in interval notation we write it like the expression below:

Domain of f(x): (-∞,3),(3,∞)

Now, what happens when you are asked to find the domain of a function under a radical?

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Remember, for the domain we are only interested in real numbers so the inside of a radical cannot be negative if we want only real numbers. Therefore, the inside of the radical has to be 0 or a positive number.

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The next step to graphing a rational function is finding x and y intercepts:

Finding X and Y Intercepts:

The x-intercept of a line is the point at which the line

crosses the x axis. ( i.e. where the y value equals 0 )

x-intercept = ( x, 0 )

The y-intercept of a line is the point at which the line

crosses the y axis. ( i.e. where the x value equals 0 )

y-intercept = ( 0, y )

Example:

image

To find the x-intercept, set y = 0 and solve for x.

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To find the y-intercept, set x = 0 and solve for y.

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Therefore, the x-intercept is ( -4, 0 ) and the y-intercept

is ( 0, 16 ).

The next step is to use all of the information you have complied and graph your function.

Graphing:

Example:

image

Step One: Check for Asymptotes

Here, this equation follows:

* If n = m, then the horizontal asymptote is equal to the ratio of the leading coefficients.

So, the horizontal asymptote is y=1.

Step Two: Determine the Domain

Because there is a bad x-guy in the denominator when x is equal to one there is a domain restriction.

So, the domain in interval notation is:

image

Step Three: Find all x and y Intercepts

To find the x-intercept, set y = 0 and solve for x.

image

To find the y-intercept, set x = 0 and solve for y.

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Therefore, the x-intercept is ( -5/2, 0 ) and the y-intercept is ( 0, -5 )

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Step Three: Now, graph your function and you are good to go!

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Rational Functions. Place of Publication Not Identified: Cerebellum Corporation, 2001. Web.

“X and Y-Intercepts.” X and Y-Intercepts. N.p., n.d. Web. 03 Apr. 2016.

My name is Gabriella Dalton and I am a sophomore at UNM. I am majoring in Applied Mathematics and Spanish. I have worked at CAPS for four semesters and currently a tutor for Algebra, Pre-Calculus, Trigonometry, and Calculus. 


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