Rational Function Graphs

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(x) and Q(x) are polynomials.

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

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

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

Compare the degrees of the numerator and denominator: If the degree of the numerator is less, the horizontal asymptote is $$y = 0$$ ; if equal, $$y = \frac{a}{b}$$ where a and b are leading coefficients; if greater, there is no horizontal asymptote.

A hole occurs in a rational function at a value of x where both the numerator and denominator are zero, indicating that the function is not defined at that point.

To find holes, factor both the numerator and denominator, and identify common factors that cancel out.

The zeros of a rational function are the values of x that make the numerator equal to zero, indicating where the function crosses the x-axis.

Set the numerator of the rational function equal to zero and solve for x.

The degree of a polynomial affects the behavior of the function, including the number of zeros, vertical and horizontal asymptotes.