09/14 9. Rational Expressions and Functions Free Online FlashCards

What is a rational expression?

A rational expression has the form $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x)\ne0$.

How are restrictions found?

Set the original denominator equal to zero, solve, and exclude those values from the domain.

How do you simplify a rational expression?

Factor completely, state the original restrictions, cancel common factors, and write the simplified expression.

How do you divide rational expressions?

To divide rational expressions, multiply the first expression by the reciprocal of the divisor: $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$.

How are rational expressions added?

Use a common denominator: $\frac{A}{D}+\frac{B}{D}=\frac{A+B}{D}$. Find the least common denominator and combine the resulting numerators.

What is the reliable method for solving rational equations?

Find restrictions, determine the LCD, multiply every term by the LCD, solve, and check each solution in the original equation.

Solve $\frac{x}{12}=\frac{5}{8}$.

Cross multiplication gives $8x=60$, so $x=7.5$.

How are rational-function intercepts found?

An $x$-intercept occurs where the reduced numerator is zero and the denominator is nonzero. The $y$-intercept is $R(0)$, if $0$ is in the domain.

What distinguishes a hole from a vertical asymptote?

A canceled denominator factor creates a hole; a denominator factor that remains after simplification creates a vertical asymptote.

How do degree comparisons determine a horizontal asymptote?

For the reduced function, if the numerator degree is less than the denominator degree, the horizontal asymptote is $y=0$. If the degrees are equal, it is the ratio of leading coefficients.

When does a rational function have a slant asymptote?

When the numerator degree is exactly one greater than the denominator degree, polynomial division gives a slant asymptote.

What is a useful strategy for graphing rational functions?

Factor first, state restrictions, locate holes and vertical asymptotes, find horizontal or slant asymptotes and intercepts, then test points in the resulting intervals.