Rational Functions Help
Introduction to Rational Functions
A rational function is a function that can be written as one polynomial divided by another.
Polynomial functions are a special kind of rational function whose denominator function is Q ( x ) = 1. While the graph of every polynomial function has exactly one y -intercept, the graph of a rational function might not have a y -intercept. If it has a y -intercept, it can be found by setting x equal to zero. If it has any x -intercepts, they can be found by setting the numerator equal to zero.
The graphs of rational functions often come in pieces. For every x -value that causes a zero in the denominator, there will be a break in the graph. If the function is reduced to lowest terms (the numerator and denominator have no common factors), then there will be a vertical asymptote at these breaks. The graph rises (or falls) very fast near these asymptotes. The graph in Figure 8.1 is the graph of . It has a vertical asymptote at the line x = 1 because x = 1 causes a zero in the denominator.
A vertical asymptote shows that the y -values get large when the x -values get close to a zero in the denominator. To see this, we will evaluate at x = 0.99 and x = 1.01, two x -values close to a zero in the denominator.
The graph flattens out horizontally near a horizontal asymptote . The graph in Figure 8.1 has the x -axis as its horizontal asymptote. A horizontal asymptote shows that as x gets very large, the y -values get very close to a fixed number. In the function , there is a horizontal asymptote at y = 0 (the x -axis). This means that as x gets large, the y -values get close to 0.
Finding the Horizontal Asymptote
Vertical asymptotes are easy to find—set the denominator equal to zero and solve for x . Whether or not a graph has a horizontal asymptote depends on the degree of the numerator and of the denominator.
- If the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote.
- If the degree of the denominator is larger than the degree of the numerator, there is a horizontal asymptote at y = 0, which is the x -axis.
- If the degree of the numerator equals the degree of the denominator, there is a horizontal asymptote at , where a n is the leading coefficient of the numerator and b m is the leading coefficient of the denominator.
Find the intercepts, vertical asymptotes, and horizontal asymptotes.
Solving 3 x + 1 = 0 we get . The vertical line is the vertical asymptote for this graph. There is no horizontal asymptote because the degree of the numerator, 2, is more than the degree of the denominator, 1. The x -intercepts are ±4 (from x 2 − 16 = 0) and the y -intercept is
When we solve x 2 − 4 x − 5 = 0, we get the solutions x = 5, − 1. This graph has two vertical asymptotes, the vertical lines x = 5 and x = − 1. The x -axis is the horizontal asymptote because the degree of the numerator, 0, is less than the degree of the denominator, 2. (A reminder, the degree of a constant term is 0, 15 = 15 x 0 .) There is no x -intercept because the numerator of this fraction is always 15, it is never 0. The y -intercept is
Because x 2 +2 = 0 has no real solutions, this graph has no vertical asymptote. There is a horizontal asymptote at because the degree of the numerator and denominator is the same. The x -intercept is 0 (from 3 x 2 = 0). The y -intercept is
The reason we can find the horizontal asymptotes so easily is that for large values of x , only the leading terms in the numerator and denominator really matter. The examples below will show an algebraic reason for the rules above. For any fixed number c any positive power on x ,
is almost 0 for large values of x . For example, in , if we let x be any large number, the fraction will be close to 0.
The larger x is, the closer to is.
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