Exploring Rational Function Graphs

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

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

To find the horizontal asymptote of a rational function $$f(x) = \frac{a_n x^n + ...}{b_m x^m + ...}$$ , compare the degrees n and m of the polynomials in the numerator and denominator. If n < m, the asymptote is y = 0; if n = m, it is y = \frac{a_n}{b_m}; if n > m, there is no horizontal asymptote.

The y-intercept of a function is the value of the function when x = 0, found by evaluating f(0).

To find the y-intercept of a rational function $$f(x) = \frac{P(x)}{Q(x)}$$ , substitute x = 0 into the function and simplify to find f(0).

End behavior describes how the values of a function behave as x approaches positive or negative infinity.

To analyze the end behavior of a rational function, determine the degrees of the numerator and denominator and use the leading coefficients to find the horizontal asymptote.

The leading coefficient of the highest degree term in the numerator and denominator helps determine the horizontal asymptote and the end behavior of the function.

At vertical asymptotes, the function approaches infinity or negative infinity, and the graph will not cross these lines.

Vertical asymptotes occur where the denominator of the rational function equals zero, provided the numerator does not also equal zero at those points.