Rational Function Transformations

A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form $$f(x) = \frac{P(x)}{Q(x)}$$ , where P and Q are polynomials.

The domain of a rational function is all real numbers except where the denominator is zero.

The range of a rational function is the set of all possible output values, which can vary based on the function's behavior and asymptotes.

A vertical asymptote is a vertical line $$x = a$$ where the function approaches infinity or negative infinity as it approaches the line.

Vertical asymptotes can be found by setting the denominator of the rational function equal to zero and solving for x.

A horizontal asymptote is a horizontal line $$y = b$$ that the graph of the function approaches as x approaches infinity or negative infinity.

Horizontal asymptotes can be found by analyzing the degrees of the numerator and denominator polynomials.

The degree of the numerator and denominator determines the behavior of the function at infinity and the existence of horizontal asymptotes.

A hole occurs in a rational function at a value of x where both the numerator and denominator are zero, indicating a removable discontinuity.

Transformations such as shifts, stretches, and reflections can change the position and shape of the graph of a rational function.