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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section GR-3 Rational Functions

Worksheet Graphing Rational Functions

For this outcome Specifically, for this outcome, students should be able to:
  • Identify the domain.
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  • Identify the vertical asymptotes.
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  • Identify the horizontal or oblique asymptotes.
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  • Identify the x-intercepts and y-intercept.
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  • Determine the behavior of the function near the asymptotes.
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  • Determine the end behavior of the function.
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  • Sketch the graph of a rational function.
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  • Identify holes (deleted points) in the graph of a rational function.
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Rational Functions
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A rational function is a function of the form \(f(x) = \frac{p(x)}{q(x)}\text{,}\) where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) \ne 0\text{.}\)
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Asymptotes
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For rational functions, there are three types of asymptotes: Vertical, Horizontal, and Oblique.
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  • Vertical asymptotes: These occur at values of \(x\) where the denominator is zero and the numerator is non-zero. The function cannot cross a vertical asymptote since this is where the function is undefined.
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  • Horizontal asymptotes: These occur when the degree of the numerator is less than or equal to the degree of the denominator. A function can cross a horizontal asymptote and this tells us the end behavior of the function. Meaning that as \(x \to \pm\infty\text{,}\) the function approaches the horizontal asymptote (the function value approaches \(y=\) a number).
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  • Oblique asymptotes: These occur when the degree of the numerator is larger than the degree of the denominator. To find an oblique asymptote, perform polynomial long division and the quotient (excluding the remainder) is the oblique asymptote. Like horizontal asymptotes, a function can cross an oblique asymptote and this tells us the end behavior of the function (whether the function value approaches \(\pm\infty\)).
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Graphing Rational Functions
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When graphing a rational function, follow these steps:
  1. Identify the domain of the function.
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  2. Factor the numerator and denominator to identify any holes in the graph (values of \(x\) where both the numerator and denominator are zero).
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  3. Find and plot any vertical asymptotes.
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  4. Find and plot any horizontal or oblique asymptotes.
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  5. Determine the x-intercepts (where the numerator is zero) and y-intercept (where x=0).
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  6. Determine the behavior of the function near each vertical asymptote.
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  7. Determine the end behavior of the function (as \(x \to \pm\infty\)).
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  8. Sketch a smooth curve that reflects all of these features.
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Worksheet GR-3 Rational Functions

Asymptotes
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Rational Functions
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Section GR-3 Rubric

Worksheet Worksheet

Before attempting the GR-3 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
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Table 16. GR3 Rubric
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Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner No algebraic errors OR some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Basics of Graphing Graph is labeled with units for each axis. The graph includes all of the important features (intercepts, asymptotes, deleted points, etc).  The graph is appropriately drawn to accommodate all features. Graph is labeled with units for each axis and includes most of the important features or the graph has all of the important features but is not labeled. Graph is not labeled and has some of the important features. No graph was given or the given graph is not labeled and does not have the correct features of the function.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution. Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution. Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
State the domain Provided a correct, simplified domain using correct interval notation. Provided the correct domain using interval notation. Provided a domain. No evidence of finding the domain.
Find the asymptotes Used appropriate factoring or division methods to find and label the vertical, horizontal and oblique asymptotes, and wrote them using correct notation. Used factoring or division methods to find and label the vertical and horizontal asymptotes with minor errors. Found some of the asymptotes. No evidence of finding the asymptotes
Find x and y intercept The correct x and y-intercepts were given using correct notation. The correct x and y-intercepts were given. Points were given that are not the intercepts. No intercepts were given.
Find deleted points Used appropriate factoring methods to find any holes; wrote the correct deleted point using correct notation. Used factoring methods to find any holes; wrote the deleted point. Gave an x-value of the hole. No evidence of finding the deleted points were given.
Rational Graph Graph contains all relevant information; has correct end-behavior, asymptotes, intercepts, deleted points, behavior at intercepts, and slopes. Graph contains most of the relevant information; has partially correct end-behavior, slopes, asymptotes, intercepts, deleted points or behavior at intercepts. Relevant parts of the graph are missing. errors in the end-behavior, slopes, asymptotes, intercepts, deleted points, or behavior at intercepts. No attempt at graphing the function was given.
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Worksheet GR-3 Rational Functions

This outcome covers rational functions. You should be able to identify the domain and range of a rational function. Be able to determine the vertical, horizontal, and oblique asymptotes of a rational function. You should also be able to find the x-intercepts and y-intercept of a rational function. If a rational function has holes (deleted points), you should be able to identify these as well. You should also be able to determine the behavior of a rational function near the asymptotes and the end behavior of a rational function. You should also be able to graph rational functions. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.
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For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
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1.

Given the function \(r(x)=\frac{4x^2}{x^2-2x-3}\text{.}\)
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(a)
State the domain.
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Hint.
Set the denominator equal to zero and solve for x to find the values that are not in the domain.
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Answer.
\((-\infty,-1)\cup(-1,3)\cup(3,\infty)\)
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Solution.
To find the domain, we set the denominator equal to zero and solve for x. We have: \(x^2-2x-3=0\text{.}\) Factoring, we get \((x-3)(x+1)=0\text{.}\) Therefore, \(x=3\) and \(x=-1\) are not in the domain. The domain is \((-\infty,-1)\cup(-1,3)\cup(3,\infty)\text{.}\)
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(b)
State any intercepts and/or asymptotes.
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Hint.
To find the vertical asymptotes, we look at the values that are not in the domain. To find the horizontal or oblique asymptote, we compare the degrees of the numerator and denominator.
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Answer.
Vertical asymptotes: x=-1 and x=3. Horizontal asymptote: y=4. x-intercept: (0,0). y-intercept: (0,0).
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Solution.
To find the vertical asymptotes, we look at the values that are not in the domain. We have vertical asymptotes at \(x=-1\) and \(x=3\text{.}\) To find the horizontal or oblique asymptote, we compare the degrees of the numerator and denominator. The degree of the numerator is 2 and the degree of the denominator is 2. Since the degrees are equal, there is a horizontal asymptote at \(y=\frac{4}{1}=4\text{.}\) To find the x-intercept, we set the numerator equal to zero and solve for x. We have \(4x^2=0\text{,}\) which gives us an x-intercept at \((0,0)\text{.}\) To find the y-intercept, we evaluate the function at x=0. We have \(r(0)=\frac{4(0)^2}{(0)^2-2(0)-3}=0\text{,}\) which gives us a y-intercept at \((0,0)\text{.}\)
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(c)
State any deleted points.
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Hint.
Deleted points occur when there is a common factor in the numerator and denominator that can be canceled.
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Answer.
No deleted points.
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(d)
Provide a sketch of the graph.
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Hint.
Use the intercepts, asymptotes, and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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2.

Given the function \(r(x)=\frac{3x-x^2}{2x-2}\text{.}\)
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(a)
State the domain.
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Hint.
Set the denominator equal to zero and solve for x to find the values that are not in the domain.
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Answer.
\((-\infty,1)\cup(1,\infty)\)
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Solution.
To find the domain, we set the denominator equal to zero and solve for x. We have: \(2x-2=0\text{.}\) Solving for x gives us \(x=1\text{.}\) Therefore, the domain is \((-\infty,1)\cup(1,\infty)\text{.}\)
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(b)
State any intercepts and/or asymptotes.
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Hint.
To find the vertical asymptotes, we look at the values that are not in the domain. To find the horizontal or oblique asymptote, we compare the degrees of the numerator and denominator.
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Answer.
Vertical asymptote: x=1. Oblique asymptote: y=-\frac{1}{2}x+1. x-intercept: (0,0). y-intercept: (0,0).
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Solution.
We have a vertical asymptote at \(x=1\text{.}\) To find the oblique asymptote, we perform polynomial long division. The result is \(y=-\frac{1}{2}x+1\text{.}\) For the intercepts, we set the numerator equal to zero to find the x-intercept and evaluate the function at x=0 to find the y-intercept.
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(c)
State any deleted points.
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Hint.
Deleted points occur when there is a common factor in the numerator and denominator that can be canceled.
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Answer.
No deleted points.
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(d)
Provide a sketch of the graph.
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Hint.
Use the intercepts, asymptotes, and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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3.

Given the function \(r(x)=\frac{x-2}{-4x^2+6x-2}\text{.}\)
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(a)
State the domain.
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Hint.
Set the denominator equal to zero and solve for x to find the values that are not in the domain.
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Answer.
\((-\infty,\frac{1}{2})\cup(\frac{1}{2},1)\cup(1,\infty)\)
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Solution.
To find the domain, we set the denominator equal to zero and solve for x. We have: \(-4x^2+6x-2=0\text{.}\) Factoring, we get \((-2x+1)(2x-2)=0\text{.}\) Therefore, \(x=\frac{1}{2}\) and \(x=1\) are not in the domain. The domain is \((-\infty,\frac{1}{2})\cup(\frac{1}{2},1)\cup(1,\infty)\text{.}\)
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(b)
State any intercepts and/or asymptotes.
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Hint.
To find the vertical asymptotes, we look at the values that are not in the domain. To find the horizontal or oblique asymptote, we compare the degrees of the numerator and denominator.
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Answer.
Vertical asymptotes: x=\frac{1}{2} and x=1. Horizontal asymptote: y=0. x-intercept: (2,0). y-intercept: (0,1).
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Solution.
We have vertical asymptotes at \(x=\frac{1}{2}\) and \(x=1\text{.}\) To find the horizontal asymptote, we can see that the degree of the numerator is less than the degree of the denominator. The result is \(y=0\text{.}\) For the intercepts, we set the numerator equal to zero to find the x-intercept and evaluate the function at x=0 to find the y-intercept.
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(c)
State any deleted points.
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Hint.
Deleted points occur when there is a common factor in the numerator and denominator that can be canceled.
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Answer.
No deleted points.
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(d)
Provide a sketch of the graph.
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Hint.
Use the intercepts, asymptotes, and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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4.

Given the function \(r(x)=\frac{-3x^2-4x+1}{x+1}\text{.}\)
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(a)
State the domain.
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Hint.
Set the denominator equal to zero and solve for x to find the values that are not in the domain.
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Answer.
\((-\infty,-1)\cup(-1,\infty)\)
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Solution.
To find the domain, we set the denominator equal to zero and solve for x. We have: \(x+1=0\text{.}\) Therefore, \(x=-1\) is not in the domain. The domain is \((-\infty,-1)\cup(-1,\infty)\text{.}\)
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(b)
State any intercepts and/or asymptotes.
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Hint.
To find the vertical asymptotes, we look at the values that are not in the domain. To find the horizontal or oblique asymptote, we compare the degrees of the numerator and denominator.
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Answer.
No vertical asymptotes. No horizontal or oblique asymptote. x-intercept: (\frac{1}{3},0). y-intercept: (0,-1).
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Solution.
There are no vertical asymptotes since the only factor in the denominator cancels out with a factor in the numerator. There are also no horizontal or oblique asymptotes because the degree of the numerator is greater than the degree of the denominator.
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(c)
State any deleted points.
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Hint.
Deleted points occur when there is a common factor in the numerator and denominator that can be canceled.
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Answer.
The function has a hole at x=-1, which corresponds to the deleted point (-1,2).
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(d)
Provide a sketch of the graph.
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Hint.
Use the intercepts, asymptotes, and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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5.

Given the function \(r(x)=\frac{x^3+3x^2+6x+8}{x^2-x-6}\text{.}\)
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(a)
State the domain.
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Hint.
Set the denominator equal to zero and solve for x to find the values that are not in the domain.
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Answer.
\((-\infty,-2)\cup(-2,3)\cup(3,\infty)\)
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Solution.
To find the domain, we set the denominator equal to zero and solve for x. We have: \(x^2-x-6=0\text{.}\) Factoring, we get \((x-3)(x+2)=0\text{.}\) Therefore, \(x=-2\) and \(x=3\) are not in the domain. The domain is \((-\infty,-2)\cup(-2,3)\cup(3,\infty)\text{.}\)
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(b)
State any intercepts and/or asymptotes.
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Hint.
To find the vertical asymptotes, we look at the values that are not in the domain. To find the horizontal or oblique asymptote, we compare the degrees of the numerator and denominator.
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Answer.
Vertical asymptote: x=3. Oblique asymptote: y=x+4. x-intercept: none. y-intercept: (0,-\frac{4}{3}).
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Solution.
We have a vertical asymptote at \(x=3\text{.}\) To find the oblique asymptote, we perform polynomial long division on the factored form of the function. The result is \(y=x+4\text{.}\) For the intercepts, we set the numerator equal to zero to find the x-intercept and evaluate the function at x=0 to find the y-intercept.
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(c)
State any deleted points.
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Hint.
Deleted points occur when there is a common factor in the numerator and denominator that can be canceled.
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Answer.
The function has a hole at x=-2, which corresponds to the point (-2,-\frac{6}{5}).
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Solution.
To find any deleted points, we look for common factors in the numerator and denominator. We can factor the denominator as \((x-3)(x+2)\text{.}\) We can factor the numerator as \((x+2)(x^2+x+4)\text{.}\) Therefore, there is a common factor of \((x+2)\) that can be canceled. This means there is a hole at x=-2. To find the y-coordinate of the hole, we can evaluate the function at x=-2 after canceling the common factor. This gives us \(r(-2)=-\frac{6}{5}\text{.}\) Therefore, the hole is at the point (-2,-\frac{6}{5}).
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(d)
Provide a sketch of the graph.
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Hint.
Use the intercepts, asymptotes, and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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