Algebra 2 Unit 5: Rational Functions — Free Review Games.
This unit covers simplifying rational expressions, asymptotes and solving rational equations — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers simplifying rational expressions, asymptotes and solving rational equations — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Simplifying Rational Expressions
A rational expression is a fraction where the numerator and denominator are polynomials. To simplify, factor both completely and cancel common factors. You must state any values excluded from the domain (values that make the original denominator zero).
Key Points
- Factor numerator and denominator fully before canceling
- Only cancel factors (multiplied terms), never terms being added or subtracted
- Excluded values come from the original denominator, not the simplified one
- A canceled factor creates a hole in the graph, not an asymptote
Simplify: (x² - 9) / (x² - x - 6)
Factor the numerator as (x+3)(x-3) and the denominator as (x-3)(x+2). Cancel the common factor (x-3) to get (x+3)/(x+2). The excluded values are x = 3 and x = -2, because both make the original denominator zero.
2 Asymptotes
Vertical asymptotes occur where the simplified denominator equals zero. Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator. You must be able to identify both types and know that holes differ from vertical asymptotes.
Key Points
- Vertical asymptote: set the simplified denominator equal to zero and solve
- If degree of numerator < degree of denominator, horizontal asymptote is y = 0
- If degrees are equal, horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator)
- If degree of numerator > degree of denominator, there is no horizontal asymptote (oblique or none)
Find all asymptotes of f(x) = (2x² + 1) / (x² - 4)
Factor the denominator as (x+2)(x-2); since neither factor cancels with the numerator, vertical asymptotes are x = 2 and x = -2. The degrees of numerator and denominator are both 2, so the horizontal asymptote is y = 2/1 = 2.
3 Solving Rational Equations
To solve a rational equation, multiply every term by the least common denominator (LCD) to eliminate all fractions. After solving, check every solution against the original excluded values and reject any that cause division by zero (extraneous solutions).
Key Points
- Find the LCD of all denominators in the equation
- Multiply both sides by the LCD to clear all fractions
- Always check solutions in the original equation to identify extraneous ones
- An extraneous solution is one that satisfies the simplified equation but is excluded from the original domain
Solve: 3/(x-2) + 1 = 5/(x-2)
Multiply every term by (x-2) to get 3 + (x-2) = 5, which simplifies to x + 1 = 5, so x = 4. Check: x = 4 does not make x-2 equal zero, so it is valid. The solution is x = 4.
Questions, answered.
What is Rational Functions?
Rational Functions is Unit 5 of Algebra 2, covering simplifying rational expressions, asymptotes and solving rational equations.
How to study for Algebra 2 Unit 5?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.