Algebra 2: Chapter 4

Asymptote

A line that a curve approaches as it heads towards infinity, indicating a limit in the behavior of the function.

Constant of Variation

The ratio of two variables that remain consistent in a direct variation or inverse variation relationship.

Inverse Variation

A relationship where one variable increases as the other decreases, defined mathematically as the product of the two variables being constant.

Reciprocal Function

A type of function defined as the inverse of a given function, resulting in a graph that contains asymptotes and has a vertical and horizontal reflection.

How are inverse variations related to the reciprocal function?

Inverse variations can be expressed using reciprocal functions, where one variable is the reciprocal of the other, leading to a constant product. This relationship highlights how changes in one variable directly affect the other.

How can you graph a rational function?

To graph a rational function, identify vertical and horizontal asymptotes, find intercepts, and plot key points while observing the behavior near asymptotes.

Rational expression

A quotient of two polynomials, where the denominator is not zero. Rational expressions can be simplified, added, subtracted, multiplied, and divided.

Rational function

A function represented by the ratio of two polynomials, where the denominator is not zero. Rational functions can have asymptotes and exhibit various behaviors as their inputs vary.

Simplified form of a rational expression

is obtained by factoring and reducing common factors in the numerator and denominator.

How does understanding operations help you multiply and divide rational functions?

Understanding operations such as factoring, simplifying, and identifying excluded values is crucial. When multiplying, you multiply the numerators and denominators, simplifying before or after. When dividing, you multiply by the reciprocal of the divisor. Both processes rely heavily on factoring to cancel out.

Compound fraction

A fraction where the numerator, the denominator, or both contain fractions.

How do you rewrite rational expressions to find sums and differences?

To find the sums and differences of rational expressions, you first need to identify the Least Common Denominator (LCD) of all the expressions. This involves factoring the denominators of each expression to find all unique factors and their highest powers. Once the LCD is determined, rewrite each rational expression by multiplying its numerator and denominator by the factors necessary to make its denominator the LCD. After all expressions share a common denominator, you can then add or subtract their numerators while keeping the common denominator. Finally, simplify the resulting rational expression if possible.