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The Graph of Rational Functions

The Graph of Rational Functions

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The Graph of Rational Functions

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Tutorial

  • Asymptotes
  • Vertical Asymptotes
  • Horizontal Asymptotes
  • Oblique Asymptotes

Asymptotes

The graphs of rational functions have asymptotes, which are lines that aren't part of the curve itself, but it dictates certain behavior about the curve.  Let's begin with a definition of asymptotes, and then we will explore three different types of asymptotes on the graphs of rational functions.

Asymptote: a line that a curve approaches; the distance between the curve and an asymptote approaches zero. 

The main characteristic of asymptotes is that the curve continues to approach the asymptote at the extreme ends of the graph.  To investigate this further, we are going to look at vertical, horizontal, and oblique asymptotes in rational functions. 

Vertical Asymptotes

Vertical asymptote: a vertical line that a curve approaches from the left or right; the curve tends towards positive or negative infinity, and its distance to the vertical line tends towards zero.

Below is a sketch of the graph of f left parenthesis x right parenthesis space equals space space fraction numerator x plus 1 over denominator x squared minus x minus 6 end fraction

The dashed vertical lines in the graph above are the locations of the vertical asymptotes.  To find vertical asymptotes, we consider x-values that make the denominator equal to zero.  These represent values for which the function is undefined, since we cannot divide by zero. 

Finding vertical asymptotes vertically, we set the denominator equal to zero and solve for x:


Before we can conclude that these are vertical asymptotes, we must also make sure that these x-values do not make the numerator equal zero as well.  If they do, then we have holes on the graph at these points, rather than vertical asymptotes.

We have confirmed vertical asymptotes at x = –2 and x = 3

Horizontal Asymptotes

Horizontal asymptote: a horizontal line that a curve approaches from above or below; the curve tends towards a constant value, and its distance to the horizontal line tends towards zero.

To find horizontal asymptotes of a rational function, we must analyze the degrees of the polynomials in the numerator and denominator of the fraction.  

In general, 

  • A rational function can be expressed as f left parenthesis x right parenthesis space equals space space fraction numerator p left parenthesis x right parenthesis over denominator q left parenthesis x right parenthesis end fraction
  • The degree of p left parenthesis x right parenthesis space is m
  • The degree of q left parenthesis x right parenthesis is n

Comparing the degrees of p(x) and q(x) can help us determine the equation to horizontal asymptotes:

  • If m < n, then the horizontal asymptote is the line y space equals space 0
  • If m = n, then the horizontal asymptote is the line y equals a over b, where a and b are the leading coefficients of the polynomials (we'll get to this later). 

Let's consider the previous rational function, as it also has a horizontal asymptote.  The equation, once again, is:

f left parenthesis x right parenthesis space equals space space fraction numerator x plus 1 over denominator x squared minus x minus 6 end fraction

The degree on the numerator is 1, and the degree of the denominator is 2.  Since the denominator has a greater degree, the horizontal asymptote is the line y = 0.  Here is the graph of the equation once more:

We can see that the line y = 0, or the x-axis of the graph, is the horizontal asymptote: the curve tends towards this horizontal line as x approaches positive and negative infinity. 

Next, let's consider the case where the degrees are equal.  In this case, the horizontal asymptotes is found by dividing the leading coefficients of the polynomials.  The leading coefficient is the coefficient of the highest degree term in the polynomial. 

Consider the function f left parenthesis x right parenthesis space equals space space fraction numerator 2 x squared plus x minus 2 over denominator x squared minus x minus 2 end fraction

Both the numerator and denominator are second degree polynomials.  This means that we divide the leading coefficients to find the horizontal asymptote.  The leading coefficients are 2 (in the numerator) and 1 (in the denominator).  Therefore, the horizontal asymptote is the line y = 2

Here is a sketch of the function.  We can see that there are some vertical asymptotes as well, but we want to focus here on the line y = 2 as the horizontal asymptote.

 

Oblique Asymptotes

With horizontal asymptotes, we consider either when the degrees of the polynomials in the numerator and denominator are equal to each other, or if the degree in the denominator is greater.   If the degree in the numerator is greater, then the rational function has an oblique asymptote.


Oblique asymptote: also called slant asymptote, a line that a curve approaches as x tends towards positive or negative infinity; it can be defined by the line y = mx+b, where m ≠ 0

When studying oblique asymptotes, more often than not, the degree in the numerator will only be 1 more than the degree in the denominator.  In these cases, the oblique asymptote is a linear equation in the form y = mx + b.  If the degree in the numerator is greater by more than 1 degree, the equation for the oblique asymptote will not be linear. 

Here is a graph of a function that has an oblique asymptote:

Notice that the asymptote is neither horizontal nor vertical.  To find the exact equation for an oblique asymptote, we do so through polynomial division.  In dividing the numerator by the denominator, we take only the polynomial portion of the quotient as the equation to the oblique asymptote.  Any remainder is not included in the equation.