Rational relationships

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This course is a part of Algebra II, a 23-course Topic series from Khan Academy.

Rational expressions are like fractions, but instead of integers in the numerator and the denominator, you have variable expressions! Learn how to work with such expressions. Namely, simplify, add, subtract, multiply, and divide them (much like fractions!). Then, solve some equations with rational expressions in them, and analyze the behavior of rational functions.

This course contains 10 segments:

Simplifying rational expressions

Learn how to simplify rational expressions by canceling factors that are shared by the numerator and the denominator. Sometimes this calls for factoring the numerator and the denominator in various ways.

Multiplying & dividing rational expressions

Learn how to multiply and divide rational expressions. You will be surprised to see how similar it is to multiplying and dividing fractions!

Adding & subtracting rational expressions

Learn how to add and subtract rational expressions. Like multiplication and division, this skill has a remarkable affinity to adding and subtracting fractions!

Nested fractions

Learn how to simplify rational expressions that contain further rational expressions within their numerators or denominators.

Solving rational equations

Learn how to solve equations that have a rational expression, or a few of those.

Direct and inverse variation

Learn about direct and inverse variation, which are two types of relationships between two quantities. Direct variation is simply a proportional relationship, but inverse variation is more complicated and interesting. Gain some experience with it before we dive deeper into the world of rational functions.

End behavior of rational functions

Learn about the ways in which rational functions behave as x approaches positive or negative infinity. This gets interesting Learn how to determine this behavior for any kind of rational function.

Discontinuities of rational functions

Learn about the ways in which rational functions behave when their denominator is equal to zero. This gets interesting when vertical asymptotes are involved! Learn how to determine this behavior for any kind of rational function.

Graphs of rational functions

Combine your knowledge of intercepts, horizontal asymptotes, vertical asymptotes, and removable discontinuities, in order to analyze entire graphs of rational functions!

Modeling with rational functions

See some examples of how rational functions and equations can come in handy when solving real-world word problems.