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Introduction to Rational Functions

A rational function is a function that can be written as one polynomial divided by another.

Rational Functions

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 Rational Functions . It has a vertical asymptote at the line x = 1 because x = 1 causes a zero in the denominator.

Rational Functions

Fig. 8.1

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 Rational Functions at x = 0.99 and x = 1.01, two x -values close to a zero in the denominator.

Rational Functions and

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 Rational Functions , 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.

Rational Functions

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.

Examples

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

Rational Functions Examples

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

Rational Functions Examples

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

Rational Functions Examples

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 ,

Rational Functions Examples

is almost 0 for large values of x . For example, in Rational Functions Examples , if we let x be any large number, the fraction will be close to 0.

Rational Functions Examples

The larger x is, the closer to Rational Functions Examples is.

Examples

From above, we know that the x -axis, or the horizontal line y = 0, is a horizontal asymptote. Here is why. Because the highest power on x is 4, we will multiply the fraction by which reduces to 1, so we are not changing the fraction.

For large values of x , 3/ x , 5/ x ² , 1/ x ³ , 6/ x ⁴ , 8/ x ² , and 1/ x ⁴ are very close to zero, so for large values of x ,
The degree of the numerator equals the degree of the denominator, so the graph of this function has a horizontal asymptote at the line y = 4/9. Here is why. Because the largest power on x is 3, we will multiply the fraction by .

For large values of Rational Functions Examples is close to .

These steps are not necessary to find the horizontal asymptotes, only the three rules earlier in this guide.

Rational Functions Practice Problems

Practice

Find the intercepts, vertical asymptotes, and horizontal asymptotes.

Rational Functions Practice

Solutions

The vertical is asymptote , from 2 x + 3 = 0. The horizontal asymptote is because the numerator and denominator have the same degree. The x -intercept is −2, from x + 2 = 0. The y -intercept is
The vertical asymptotes are x = − 5 and x = 4, from x ² + x − 20 = 0. The horizontal asymptote is y = 0 because the denominator has the higher degree. The x -intercept is 0, from −3 x = 0. The y -intercept is
There is no vertical asymptote because x ² + 1 = 0 has no real solution. The horizontal asymptote is y = 1/1 = 1 because the numerator and denominator have the same degree. The x -intercepts are ±1, from x ² − 1 = 0. The y -intercept is
The vertical asymptote is from 8 x + 3 = 0. There is no horizontal asymptote because the numerator has the higher degree. The x -intercepts are , from 9 x ² − 1 = 0. The y -intercept is
There is no vertical asymptote because x ² + 4 = 0 has no real solution. There is no horizontal asymptote because the numerator has the higher degree. The x -intercept is −1, from x ³ + 1 = 0. The y -intercept is
The vertical asymptote is x = 0, from x ² = 0. The horizontal asymptote y = 0 because the denominator has the higher degree. There is no x -intercept because the numerator is 2, never 0. There is no y -intercept because 2/0 ² is not defined.

Practice problems for this concept can be found at: Rational Functions Practice Test.

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