Questions and Answers
What is a rational expression?
What does it mean to simplify rational expressions?
Remove common factors from the numerator and denominator, state restrictions, and simplify.
A rational expression can have a denominator equal to zero.
False
What are the steps for multiplying rational expressions?
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What is the first step in dividing rational expressions?
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What are the steps for solving rational equations?
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What is direct variation?
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In inverse variation, y varies directly with x.
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What is the constant of variation in direct and inverse variations?
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What is the first step in graphing simple rational functions?
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Which statement is true regarding the horizontal asymptote (HA) of a rational function?
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How do you find the x-intercept of a rational function?
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What is joint variation?
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In combined variations, how does z vary with respect to x, y, and w?
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What should you do to graph rational functions effectively?
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Study Notes
Rational Expressions
- Defined as an expression in the form of p(x)/q(x) where p(x) and q(x) are polynomials.
Simplifying Rational Expressions
- Remove common factors from the numerator and denominator.
- Identify restrictions to avoid zero in the denominator.
Multiplying Rational Expressions
- Factor all numerators and denominators before multiplication.
- State restrictions and simplify common factors.
Dividing Rational Expressions
- Factor both numerators and denominators.
- Flip the second fraction before proceeding with multiplication.
- Identify restrictions from both the original and flipped denominators.
Adding and Subtracting Rational Expressions
- Determine the least common denominator (LCD) as the highest degree of factors from all denominators.
- Multiply the entire numerator and denominator by the LCD.
- Perform addition or subtraction on the numerators and simplify.
Complex Fractions
- Find the LCD of all involved fractions.
- Multiply numerator and denominator by the LCD, remembering to factor everything first.
Solving Rational Equations
- Find the least common denominator (LCD) and multiply both sides by it.
- Simplify the equation and solve, checking for extraneous solutions.
Solving Proportions
- Ensure there is one fraction on each side before solving.
- Use cross multiplication to solve the equation, checking for extraneous solutions.
Direct Variation
- Represented by the equation y = kx where k ≠ 0.
- Can be expressed as y/x = k, indicating that y changes directly with x.
Inverse Variation
- Expressed by the equation xy = k where k ≠ 0.
- Can also be rearranged to y = k/x, indicating that y varies inversely with x.
Joint Variation
- A relationship where one variable varies directly with two or more variables.
- Example: z varies jointly with x and y, resulting in z = kxy.
Combined Variations
- Combines direct and inverse variations: e.g., z = ky/x signifies direct variation with y and inverse with x.
- Another form could be z = kxy/w², showing joint variation with x and y and inverse with w².
Constant of Variation
- The ratio in direct variation or the product in inverse variation.
Graphing Simple Rational Functions
- General form: f(x) = a/(x+h) + k.
- Vertical asymptote located at h and horizontal asymptote at k.
- Graphing process involves determining asymptotes, plotting points, and analyzing domain, range, and end behavior.
Important Rules for Rational Functions
- A rational function in standard form is f(x) = p(x)/q(x).
- Horizontal asymptote rules:
- Bottom heavy (q(x) > p(x)): HA is y = 0.
- Balanced (p(x) = q(x)): HA is y = (lead coefficient of p)/(lead coefficient of q).
- Top heavy (p(x) > q(x)): No HA, may have slant asymptote.
- Vertical asymptotes identified by unique factors of the denominator, while holes arise from common factors.
Simple Rules for Rational Functions
- Factors provide insight into vertical asymptotes, x-intercepts, and holes.
- To find x-intercepts, set f(x) to zero; for y-intercepts, compute f(0).
Graphing Rational Functions in Standard Form
- Steps involve finding vertical/horizontal asymptotes, intercepts, and then graphing while considering sign tables for accuracy.
Writing Rational Functions in Standard Form
- Involves constructing functions rooted in the definitions and procedures outlined for understanding and simplifying rational expressions.
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Description
This set of flashcards covers key concepts from Algebra 2 Unit 6, focusing on rational expressions. Learn the definitions, simplification techniques, and multiplication of rational expressions to enhance your understanding of the topic.