Just as in the case of rational numbers, when speaking of rational functions, the word ‘rational’ refers to a ratio. In the case of rational functions, it refers to the ratio of two polynomials.
A rational function is a ratio of two polynomials.
For example, let f ( x ) = 1 / x . The numerator is a zero degree polynomial and the denominator is a first degree polynomial.
Sketch the graph of f ( x ) = 1 / x for values of x between - 2 and + 2, exclusive of 0.
Notice that as one picks input values x closer to 0, the output values y get further from 0 in size. For this particular function, the graph ‘approaches’ the y-axis as x approaches 0. The y-axis is a vertical asymptote of the graph.
Now sketch the graph for values of x between 2 and 10 and between -1 and -2.
Notice that as one picks input values further from 0, the output values are closer to 0 in size. The x-axis is a horizontal asymptote of the graph.
Rational functions may have vertical and horizontal asymptotes.
Assuming that the numerator and denominator of a rational function have no common factors, the vertical asymptotes occur at the zeros of the denominator.
In the example above, 0 is the zero of the denominator, so the vertical asymptote occurs at x = 0. In fact, x = 0 is the equation of the vertical asymptote. [Remember that vertical lines have equations of the form x = c, where c is some constant.]
Write the equations of the vertical asymptotes of f ( x ) = 1 / ( x2 - 1 ).
In order to find the horizontal asymptote of a rational function (there can be at most one), it is necessary to find the quotient of the ratio. For example, if f ( x ) = 1 / x, the quotient is 0 and the remainder is 1. This means that x goes into 1 zero times with a remainder of 1. Thus y = 0 is the horizontal asymptote. More generally, it is the quotient asymptote. But in this case, the equation of y = quotient is the equation of a horizontal line, so it is called a horizontal asymptote.
Find the equations and sketch the graphs of of the vertical and horizontal asymptotes of the polynomial function f ( x ) = ( 2x2 - 8 ) / ( x2 - 1 ).
If the numerator and denominator of a rational function contain no common factors, then the x-intercepts occur at the zeros of the numerator.
The x-intercepts will be either transitive or intransitive according to the same principle which applies to polynomial functions: If c is a zero of the numerator and is of odd multiplicity, then the point ( 0, c ) is a transitive x-intercept. If c is a zero of the numerator and is of even multiplicity, then the point ( 0, c ) is an intransitive x-intercept. [Recall that the graph crosses the x-axis at a transitive x-intercept, and intersects but does not cross the x-axis at an intransitive x-intercept.]
The graph may or may not cross the horizontal ( quotient ) asymptote.
The points of intersection of a graph and its horizontal
( quotient ) asymptote are called the horizontal ( quotient ) intercepts.
The quotient intercepts of a polynomial function occur at the zeros of the remainder.
The quotient and remainder are both found by dividing the denominator into the numerator using the process of long division of polynomials. The quotient intercepts may be either transitive or intransitive depending upon whether the zeros of the remainder are of odd or even multiplicity.
Let f ( x ) = ( 2 x2 + x ) / ( x2 + 1 ).
Find all vertical and horizontal asymptotes and sketch their graphs. Find all x and y and quotient intercepts. Sketch the graph of f.
Sketch the graph of the polynomial function. Indicate all asymptotes and intercepts. Plot additional points only when necessary to resolve ambiguities in the graph.
f ( x ) = ( x2 - 3 ) / ( x2 - x - 2 )