Algebra 2 Unit 6 Flashcards

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Questions and Answers

What is a rational expression?

An expression of the form p(x)/q(x) where p(x) and q(x) are polynomials (correct)

An expression that contains only variables

An expression involving only integers

An expression that cannot be simplified

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?

Factor all numerators and denominators, state restrictions, simplify common factors, and simplify the expression. Signup and view all the answers

What is the first step in dividing rational expressions?

Factor all of the numerators and denominators. Signup and view all the answers

What are the steps for solving rational equations?

Find the least common denominator, multiply each side by the LCD, simplify, solve the equation, and check for extraneous solutions. Signup and view all the answers

What is direct variation?

A linear function defined by an equation of the form y = kx, where k does not equal 0. Signup and view all the answers

In inverse variation, y varies directly with x.

False Signup and view all the answers

What is the constant of variation in direct and inverse variations?

It is the ratio of the two variables in a direct variation and the product of the two variables in an inverse variation. Signup and view all the answers

What is the first step in graphing simple rational functions?

Find the vertical asymptote Signup and view all the answers

Which statement is true regarding the horizontal asymptote (HA) of a rational function?

If the degree of q(x) is greater than p(x), then HA is y = 0 Signup and view all the answers

How do you find the x-intercept of a rational function?

Set f(x) = 0 and solve for x. Signup and view all the answers

What is joint variation?

A relation where one variable varies directly with respect to each of two or more variables. Signup and view all the answers

In combined variations, how does z vary with respect to x, y, and w?

It varies directly with both x and y, and inversely with w Signup and view all the answers

What should you do to graph rational functions effectively?

Find the vertical asymptote, horizontal asymptote, intercepts, and then graph using a sign table if needed. Signup and view all the answers

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.