Algebra 2 · Precalculus · Rational expressions · Quadratic worksheets · Polynomial worksheets
Rational Expressions and Rational Functions Worksheets
Nine free guides with practice problems, answer keys, and printable PDFs
A nine guide unit on rational expressions and rational functions for Algebra II and Precalculus, listed in a recommended learning order. The guides hold 201 core practice problems with answer keys, three problem challenge sets in six of the guides, 14 interactive tools, and 19 printable PDFs. No sign up is needed.
19 printable PDFs, 91 pages. Worksheets, answer keys, and one page cheat sheets for the nine guides.
Factoring is used throughout these guides. To review factoring trinomials and differences of squares first, use the factoring polynomials flowchart guide.
Part 1: Rational expressions worksheets
Five guides on working with a single rational expression: where it is undefined, how to simplify it, and how to multiply, divide, add, and subtract.
Domain restrictions and excluded values worksheet
Find where an expression is undefined before you do any algebra. Covers zero denominators, even roots, combined restrictions, and writing the domain as an inequality, in interval notation, and in set-builder notation.
24 practice problems plus a 3 problem challenge set, with answers. Interactive tools: Domain Finder, Domain Notation Drill.
Find the excluded valuesstart fraction x plus 2 over x squared minus 9 end fractionSimplifying rational expressions worksheet
Factor the numerator and the denominator, cancel the common factors, and keep every restriction from the original denominator.
18 practice problems with answers. Interactive tool: Rational Expression Practice Generator.
Simplifystart fraction x squared minus 9 over x squared plus x minus 6 end fractionMultiplying and dividing rational expressions worksheet
Factor first, then cancel across the product. For a division, multiply by the reciprocal and record the extra restriction that comes from the divisor's numerator.
24 practice problems with answers. Interactive tool: Step by step solver.
Dividestart fraction x squared minus 4 over x plus 3 end fraction divided by start fraction x minus 2 over x squared plus 3 x end fractionAdding and subtracting rational expressions worksheet
Build the least common denominator from the factors of each denominator, rewrite each fraction, and put the second numerator in parentheses when you subtract.
20 practice problems plus a 3 problem challenge set, with answers. Interactive tool: Step by step animation.
Subtractstart fraction 3 over x minus 2 end fraction minus start fraction x plus 1 over x squared minus 4 end fractionComplex rational expressions worksheet
Simplify a fraction that contains fractions by two methods: combine each level and divide, or multiply both levels by the LCD. Track the restrictions that come from the top level, the bottom level, and the whole bottom expression.
24 practice problems plus a 3 problem challenge set, with answers. Interactive tools: Two Method Solver, Restriction Finder drill.
Simplifystart fraction 1 plus start fraction 1 over x end fraction over 1 minus start fraction 1 over x squared end fraction end fraction
Part 2: Rational equations worksheets
Two guides on solving equations that contain rational expressions and on writing those equations from a word problem.
Rational equations worksheet and extraneous solutions guide
List the excluded values, clear the denominators, and solve. Then compare each candidate with the excluded values and reject the ones that match.
25 practice problems plus a 3 problem challenge set, with answers. Interactive tools: Extraneous root visualizer, Excluded value drill.
Solvestart fraction x over x minus 4 end fraction equals start fraction 4 over x minus 4 end fraction plus xRational equation word problems worksheet
Turn work rate, travel, and shared cost problems into equations with a setup table. Then check each answer against the excluded values and against the situation.
24 practice problems plus a 3 problem challenge set, with answers. Interactive tools: Work Rate Simulator, Equation Setup Trainer.
Two pipes fill a tank in 6 hours and in 3 hours. How long together?start fraction 1 over 6 end fraction plus start fraction 1 over 3 end fraction equals start fraction 1 over t end fraction
Part 3: Rational functions worksheets
Two guides on reading the features of a rational function from its formula and turning them into a graph.
Asymptotes, holes, and intercepts of rational functions worksheet
Factor the function, then decide what each denominator factor produces: a hole when the factor cancels completely, a vertical asymptote when a copy remains. Then find the horizontal or slant asymptote and the intercepts.
18 practice problems with answers. Two visual guides: Factor Fate Map and a four graph atlas.
Find the hole and the asymptotesstart fraction x squared minus 4 over x squared minus x minus 2 end fractionDashed lines: asymptotes. Dots: intercepts. Open circle: hole.Graphing rational functions worksheet
Put the features together into a full sketch. The guide uses a six step routine and a sign chart to place each piece of the graph above or below the x-axis.
24 practice problems plus a 3 problem challenge set, with answers. Interactive tools: Rational function feature finder, Sign chart builder, Build a function from its features.
Graph the functionstart fraction x squared minus 1 over x minus 2 end fractionDashed lines: asymptotes. Dots: intercepts.
What the unit covers
A rational expression is a fraction whose numerator and denominator are polynomials, such as start fraction x plus 2 over x squared minus 9 end fraction. Any value of x that makes the denominator zero is excluded. A rational function is a function whose rule is a rational expression, and a rational equation is an equation that contains one or more rational expressions, such as start fraction x over x minus 4 end fraction equals start fraction 4 over x minus 4 end fraction plus x.
The guides are listed in a recommended learning order. Simplifying depends on factoring and on excluded values. Adding and subtracting depends on the least common denominator, which is built from the same factors. Solving a rational equation uses that denominator again and ends with a check against the excluded values. Graphing draws on factoring and excluded values, to separate holes from vertical asymptotes, and it can be studied before the two equations guides.
You do not have to start at guide 1. If you are not sure which guide matches your homework, compare your problem with the sample problem beside each title, or take the short check below.
Find your starting point
This check has nine multiple choice questions, one for each guide, in the order the guides are listed. Work each problem on paper before you choose. Every wrong answer comes from a specific error, and the feedback names it. At the end you get a list of the guides to review and the one to start with.
Nine questions. No sign up, and nothing is saved or sent.
For teachers: standards and classroom use
The table lists the Common Core standard each guide addresses. A plus sign marks standards that the Common Core designates as additional mathematics for students preparing for advanced courses.
| Guide | What students do | Standard |
|---|---|---|
| 1 Domain restrictions and excluded values | Relate the domain of a function to its formula, its graph, and the situation it describes | HSF.IF.B.5 |
| 2 Simplifying rational expressions | Use the structure of an expression to factor it and rewrite it in a simpler form | HSA.SSE.A.2 |
| 3 Multiplying and dividing rational expressions | Multiply and divide rational expressions | HSA.APR.D.7 (+) |
| 4 Adding and subtracting rational expressions | Add and subtract rational expressions with unlike denominators | HSA.APR.D.7 (+) |
| 5 Complex rational expressions | Rewrite a quotient of rational expressions as a single rational expression | HSA.APR.D.7 (+) |
| 6 Rational equations and extraneous solutions | Solve rational equations in one variable and show how extraneous solutions arise | HSA.REI.A.2 |
| 7 Rational equation word problems | Create rational equations in one variable and use them to solve problems | HSA.CED.A.1 |
| 8 Asymptotes, holes, and intercepts | Identify zeros, holes, and asymptotes of a rational function from its factored form | HSF.IF.C.7d (+) |
| 9 Graphing rational functions | Graph a rational function, showing zeros, asymptotes, and end behavior, and use polynomial division to find a slant asymptote | HSF.IF.C.7d (+) HSA.APR.D.6 |
Using the unit in class
- The starting point check works as a short pre-assessment. Each question maps to one guide, so a missed question points to one worksheet.
- Eight of the nine guides include a section on specific errors and how to correct them, which can be used for error analysis.
- Every worksheet PDF comes with an answer key, and each cheat sheet fits on one page.
The PDFs are free to copy for your classroom or your own kitchen table. If you share them, please link to this page or to the guide rather than rehosting the files, so students land on the current version.
Teachers who use these worksheets are welcome to send corrections or notes on classroom use to burketutoringinfremont@outlook.com.
Printable worksheet PDFs with answer keys
The unit has 19 PDFs, 91 pages in total. Download them together as one ZIP file, or open the list to choose single files.
Choose single PDFs
| Guide | Files |
|---|---|
| 1 Domain restrictions and excluded values |
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| 2 Simplifying rational expressions |
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| 3 Multiplying and dividing rational expressions |
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| 4 Adding and subtracting rational expressions |
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| 5 Complex rational expressions |
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| 6 Rational equations and extraneous solutions |
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| 7 Rational equation word problems |
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| 8 Asymptotes, holes, and intercepts |
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| 9 Graphing rational functions |
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About these worksheets
The problems in these guides are adapted from problem types I use in Algebra II and Precalculus tutoring sessions. Each guide page lists its own publication date and the sources it cites. To report an error in a problem or an answer key, email burketutoringinfremont@outlook.com or call or text (510) 453-0350.
About the author
Berke Sahbazoglu, founder and tutor, Burke Tutoring in Fremont, CA
I have more than ten years of tutoring experience and teach Algebra I, Algebra II, and Precalculus in homes across Fremont, Newark, and Union City. I hold a BA in Biochemistry from Washington University in St. Louis and an MS in Bioinformatics from the University of Maryland Global Campus.
Working through rational expressions with a tutor
Burke Tutoring provides in-home math tutoring in Fremont, Newark, and Union City for Algebra I, Algebra II, and Precalculus.
(510) 453-0350Serving Fremont, Newark, and Union City. See rates and formats.
Standards and further reading
- Common Core State Standards, High School Algebra: Arithmetic with Polynomials and Rational Expressions, the domain that contains the standards on rewriting rational expressions.
- California Department of Education, California Common Core State Standards: Mathematics, the edition of the standards adopted in California.
- Achieve the Core, Coherence Map, a tool that shows the connections between the Common Core mathematics standards.
- College Board, AP Precalculus course page. Unit 1 of that course is Polynomial and Rational Functions.