Rational Functions Asymptotes

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

Vertical asymptotes occur at values of $$x$$ that make the denominator zero, provided the numerator is not also zero at those points.

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

To find horizontal asymptotes, compare the degrees of the numerator and denominator: If degree of P < degree of Q, then y = 0; if degree of P = degree of Q, then y = leading coefficient of P / leading coefficient of Q; if degree of P > degree of Q, there is no horizontal asymptote.

Intercepts are points where the graph of the function crosses the axes: x-intercepts occur where $$f(x) = 0$$ and y-intercepts occur where $$x = 0$$ .

To find x-intercepts, set the numerator equal to zero and solve for $$x$$ .

To find the y-intercept, evaluate the function at $$x = 0$$ .

A function is undefined at a point if the denominator equals zero at that point, leading to division by zero.

The leading coefficient of the numerator and denominator polynomials determines the value of the horizontal asymptote when the degrees of the polynomials are equal.