Title

College Algebra
Tutorial 40:
Graphs of Rational Functions

Learning Objectives

 After completing this tutorial, you should be able to: Find the domain of a rational function. Find the vertical asymptote(s) of a rational function. Find the horizontal asymptote of a rational function. Find the oblique or slant asymptote of a rational function. Graph a rational function.

Introduction

 In this tutorial we will be looking at several aspects of rational functions.  First we will revisit the concept of domain.  On rational functions, we need to be careful that we don't use values of x that cause our denominator to be zero.  If you need a review on domain, feel free to go to Tutorial 30: Introductions to Functions.  Next, we look at vertical, horizontal and slant asymptotes.  Basically an asymptote is an imaginary line that the curve of the function gets very close to or approaches.   In the end, we put it all together and graph rational functions.  Sounds like fun, you better get to it!!!

Tutorial

 Review on Domain

 The domain is the set of all input values to which the rule applies.  These are called your independent variables. These are the values that correspond to the first components of the ordered pairs it is associated with. If you need a review on domain, feel free to go to Tutorial 30: Introductions to Functions.

 Example 1: Give the domain of the function  .

 Our restriction here is that the denominator of a fraction can never be equal to 0.  So to find our domain, we want to set the denominator equal to 0 and restrict those values.

 *The den. cannot equal to 0 *Factoring to help "solve" *1/2 and -3 are restricted values

 Our domain is all real numbers except 1/2 and -3, because 1/2 and -3 both make the denominator equal to 0, which would not give us a real number answer for our function.

 Vertical Asymptote Let     be written in lowest terms and  P and Q  are polynomial functions. If or as , then the vertical line x = a is a vertical asymptote. The line x = a is a vertical asymptote of the graph of f  if and only if the denominator Q(a) = 0  and the numerator .

In other words, you find the vertical asymptote by locating where the function is undefined.

In this case that is where the simplified rational function’s denominator is equal to 0.

Some things to note:

 You can have zero or many vertical asymptotes.  It will be x = whatever number(s) cause the denominator to be zero after you have simplified the function. You draw a vertical asymptote on the graph by putting a dashed vertical (up and down) line going through x = a as shown below.  This is where the function is undefined, so there will be NO point on the vertical asymptote itself.  The graph will approach it from both sides, but never cross over it. Below is an example of a vertical asymptote of x = 2:

 Example 2: Find the vertical asymptote of the function .

 First we want to check and see if this rational function will reduce down:

 *Factor the function

 Nothing is able to cancel out, so now we want to find where the denominator is equal to 0:

 *Set den = 0 *Set the first factor = 0 *Set the second factor = 0

 There are two vertical asymptotes: x = -3 and x = 3.

 Example 3: Find the vertical asymptote of the function .

 First we want to check and see if this rational function will reduce down:

 *Factor the function *Cancel out the common factor of x + 1

 Note how the factor x + 1 canceled out, so now we want to find where the new denominator is equal to 0:

 *Set den = 0

 There is one vertical asymptote: x = -2.

 Horizontal Asymptote Let   be written in lowest terms, where P and Q are polynomial functions and . If   as    or , then the horizontal line y = a is a horizontal asymptote.

If there is a horizontal asymptote, it will fit into one of the two following cases:

 Let   be written in lowest terms, Case I If the degree of P(x) < the degree of Q(x), then there is a horizontal asymptote at y = 0 (x-axis). Case II If the degree of P(x) = the degree of Q(x), then there is a horizontal asymptote at   .  In other words, it would be the ratio between the leading coefficient of the numerator and the leading coefficient of the denominator.

Some things to note:

 You may have 0 or 1 horizontal asymptote,  but no more than that. The graph may cross the horizontal asymptote, but it levels off and approaches it as x goes to infinity. You draw a horizontal asymptote on the graph by putting a dashed horizontal (left and right) line going through y = a. Below is an example of a horizontal asymptote of y = 3:

 Example 4: Find the horizontal asymptote of the function .

 First we want to check and see if this rational function will reduce down:

 *Factor the function

 Nothing was able to cancel out, so now we want to compare the degrees of the numerator and the denominator.  What is the degree of the numerator?  If you said 1, you are correct.  The leading term is 3x and its degree is 1. What is the degree of the denominator?  If you said 2, you are correct.  The leading term is   and its degree is 2. If you need a review of finding the degree of a polynomial, feel free to go to Tutorial 6: Polynomials. Since the degree of the numerator is 1 is less than the degree of the denominator, then there is a  horizontal asymptote at y = 0.

 Example 5: Find the horizontal asymptote of the function .

 First we want to check and see if this rational function will reduce down:

 *Factor the function *Cancel out the common factor of x + 1

 Note how the factor x + 1 canceled out, so now we want to compare the degrees of the numerator and the denominator that is left.  What is the degree of the numerator that is left?  If you said 1, you are correct.  The leading term is x and its degree is 1. What is the degree of the denominator that is left?  If you said 1, you are correct.  The leading term is x and its degree is 1. If you need a review of finding the degree of a polynomial, feel free to go to Tutorial 6: Polynomials. Since the degree of the numerator is equal to the degree of the denominator, then there is a  horizontal asymptote at

 Oblique or Slant Asymptote An oblique or a slant asymptote is an asymptote that is neither vertical or horizontal. If the degree of the numerator is one more than the degree of the denominator, then the graph of the rational function will have a slant asymptote.

Some things to note:

 The slant asymptote is the quotient part of the answer you get when you divide the numerator by the denominator. If you need a review of long division, feel free to go to Tutorial 36: Long Division. You may have 0 or 1 slant asymptote,  but no more than that. A graph can have both a vertical and a slant asymptote, but it CANNOT have both a horizontal and slant asymptote. You draw a slant asymptote on the graph by putting a dashed horizontal (left and right) line going through y = mx + b. Below is an example of a slant asymptote of y = x + 1:

 Example 6: Find the oblique asymptote of the function .

 Note that this rational function is already reduced down. Applying long division to this problem we get:

 The answer to the long division would be  . The equation for the slant asymptote is the quotient part of the answer which would be  .

 Graphing Rational Functions

 Step 1: Reduce the rational function to lowest terms and check for any open holes in the graph.

 If any factors are TOTALLY removed from the denominator, then there will not be a vertical asymptote through that value, but an open hole at that point. If this is the case, plug in the x value that causes that removed factor to be zero into the reduced rational function.  Plot this point as an open hole.

 Step 2: Find all of the asymptotes and draw them as dashed lines.

Let   be a rational function reduced to lowest terms and Q(x) has a degree of at least 1:

 There is a vertical asymptote for every root of   . There is a horizontal asymptote of y = 0 (x-axis) if the degree of P(x) < the degree of Q(x). There is a horizontal asymptote of   if the degree of P(x) = the degree of Q(x). There is an oblique or slant asymptote if the degree of P(x) is one degree higher than Q(x).  If this is the case the oblique asymptote is the quotient part of the division.

Note that a graph can have both a vertical and a slant asymptote, or both a vertical and horizontal asymptote, but it CANNOT have both a horizontal and slant asymptote.

 Step 3: Determine the symmetry.

 The graph is symmetric about the y-axis if the function is even. The graph is symmetric about the origin if the function is odd. If you need a review of even and odd functions, feel free to go to Tutorial 32: Graphs of Functions Part II.

 Step 4: Find and plot any intercepts that exist.

 The x-intercept is where the graph crosses the x-axis.  You can find this by setting y = 0 and solving for x. The y-intercept is where the graph crosses the y-axis.  You can find this by setting x = 0 and solving for y. If you need a review on intercepts, feel free to go to Tutorial 26: Equations of Lines.

 Step 5: Find and plot several other points on the graph.

 You should have AT LEAST two points in each section of the graph that is marked off by the vertical asymptotes.

 Step 6: Draw curves through the points, approaching the asymptotes.

 Note that your graph can cross over a horizontal or oblique asymptote, but it can NEVER cross over a vertical asymptote.

 Example 7: Sketch the graph of the function .

 Step 1: Reduce the rational function to lowest terms and check for any open holes in the graph.

 This function cannot be reduced any further.  This means that there will be no open holes on this graph.

 Step 2: Find all of the asymptotes and draw them as dashed lines.

 Vertical Asymptote: So now we want to find where the denominator is equal to 0:

 *Set den = 0

 There is one vertical asymptote: x = 0.

 Horizontal Asymptote: So now we want to compare the degrees of the numerator and the denominator.  What is the degree of the numerator?  If you said 0, you are correct.  The leading term is the constant -1 and its degree is 0. What is the degree of the denominator?  If you said 2, you are correct.  The leading term is   and its degree is 2. If you need a review of finding the degree of a polynomial, feel free to go to Tutorial 6: Polynomials. Since the degree of the numerator is less than the degree of the denominator, then there is a  horizontal asymptote at y = 0.

 Slant Asymptote: Since the degree of the numerator is NOT one degree higher than the degree of the denominator, there is not slant asymptote.

 Step 3: Determine the symmetry.

 Since  , the function is even.  This means the graph is symmetric about the y-axis. If you need a review of even and odd functions, feel free to go to Tutorial 32: Graphs of Functions Part II.

 Step 4: Find and plot any intercepts that exist.

 x-intercept: What value are we going to use for y?  You are correct if you said 0.

 *Plug in 0 for y *No x-intercept

 This means there is NO x-intercept.

 y-intercept: What value are we going to use for x?  You are correct if you said 0.

 *Plug in 0 for x *No y-intercept

 This means there is NO y-intercept.

 Step 5: Find and plot several other points on the graph.

 So far we have not found any points to plot on the graph.  Note how the vertical asymptote sections the graph into two parts.  I’m going to plug in two x values that are to the left of x = 0 and two that are to the right of x = 0.

 Plugging in -2 for x we get: (-2, -1/4)

 Plugging in -1 for x we get: (-1, -1)

 Plugging in 1 for x we get: (1, -1)

 Plugging in 2 for x we get: (2, -1/4)

 Step 6: Draw curves through the points, approaching the asymptotes.

 Example 8: Sketch the graph of the function .

 Step 1: Reduce the rational function to lowest terms and check for any open holes in the graph.

 *Factor the denominator

 This function cannot be reduced any further.  This means that there will be no open holes on this graph.

 Step 2: Find all of the asymptotes and draw them as dashed lines.

 Vertical Asymptote: So now we want to find where the denominator is equal to 0:

 *Set den = 0 *Set the 1st factor = 0 *Set the 2nd factor = 0

 There are two vertical asymptotes: x = -2 and x = 1.

 Horizontal Asymptote: So now we want to compare the degrees of the numerator and the denominator.  What is the degree of the numerator?  If you said 2, you are correct.  The leading term is  and its degree is 2. What is the degree of the denominator?  If you said 2, you are correct.  The leading term is  and its degree is 2. If you need a review of finding the degree of a polynomial, feel free to go to Tutorial 6: Polynomials. Since the degree of the numerator is equal to the degree of the denominator, then there is a  horizontal asymptote at

 Slant Asymptote: Since the degree of the numerator is NOT one degree higher than the degree of the denominator, there is not slant asymptote.

 Step 3: Determine the symmetry.

 Since  , the function is neither even nor odd.  If you need a review of even and odd functions, feel free to go to Tutorial 32: Graphs of Functions Part II.

 Step 4: Find and plot any intercepts that exist.

 x-intercept: What value are we going to use for y?  You are correct if you said 0.

 *Plug in 0 for y *Mult. both side by the LCD (x + 2)(x - 1) *Since x squared CANNOT be negative, there is no x-intercept

 This means there is NO x-intercept.

 y-intercept: What value are we going to use for x?  You are correct if you said 0.

 *Plug in 0 for x *y-intercept of -1/2

 The y-intercept is (0, -1/2)

 Step 5: Find and plot several other points on the graph.

 Note how the vertical asymptotes section the graph into three parts.  I’m going to plug in two x values that are to the left of x = -2, one in between x = -2 and x = 1, and two that are to the right of x = 1.

 Plugging in -4 for x we get: (-4, 17/10)

 Plugging in -3 for x we get: (-3, 5/2)

 Plugging in -1 for x we get: (-1, -1)

 Plugging in 2 for x we get: (2, 3/4)

 Plugging in 3 for x we get: (3, 1)

 Step 6: Draw curves through the points, approaching the asymptotes.

 Example 9: Sketch the graph of the function .

 Step 1: Reduce the rational function to lowest terms and check for any open holes in the graph.

 *Factor *Cancel the common factor of x

 The factor of x canceled out and there were no factors of x left in the denominator.  This means there is an open hole on the graph at x = 0. x = 0:

 *Plug in a 0 for x

 There is an open hole at (0, -4/3).

 Step 2: Find all of the asymptotes and draw them as dashed lines.

 Vertical Asymptote: So now we want to find where the new denominator is equal to 0:

 *Set den. = 0

 There is one vertical asymptote: x = -3.

 Horizontal Asymptote: Since the degree of the numerator is one degree higher than the degree of the denominator, there is a slant asymptote and no horizontal asymptote.

 Slant Asymptote: Applying long division to this problem we get: The answer to the long division would be   . The equation for the slant asymptote is the quotient part of the answer which would be  .

 Step 3: Determine the symmetry.

 Since  , the function is neither even nor odd.  If you need a review of even and odd functions, feel free to go to Tutorial 32: Graphs of Functions Part II.

 Step 4: Find and plot any intercepts that exist.

 x-intercept: What value are we going to use for y?  You are correct if you said 0.

 *Plug in 0 for y *Mult. both side by the LCD (x + 3) *Factor *Set 1st factor = 0 *Set 2nd factor = 0

 There are two x-intercepts: (4, 0) and (-1, 0).

 y-intercept: What value are we going to use for x?  You are correct if you said 0.

 *Plug in 0 for x *y-intercept of -4/3

 The y-intercept is (0, -4/3).  Note that this will be open hole, as found in step 1.

 Step 5: Find and plot several other points on the graph.

 Note how the vertical asymptote sections the graph into two parts.  I’m going to plug in two x values that are to the left of x = -3.

 Plugging in -5 for x we get: (-5, -18)

 Plugging in -4 for x we get: (-4, -24)

 Step 6: Draw curves through the points, approaching the asymptotes.

Practice Problems

 These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problem 1a:  Give the domain of the given function.

Practice Problems 2a - 2b: Find the vertical and horizontal asymptotes for the given functions.

Practice Problem 3a: Find the oblique asymptote for the given function.

Practice Problems 4a - 4b: Sketch the graph of the given function.

Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page.

 http://www.purplemath.com/modules/grphrtnl.htm This website helps you with graphing rational functions. http://www.purplemath.com/modules/grphrtnl2.htm This website helps you with graphing rational functions. http://www.purplemath.com/modules/grphrtnl3.htm This website helps you with graphing rational functions.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on March 18, 2011 by Kim Seward.