In this section we will learn about the graphs of rational functions. It is important to be able to find the asymptotes of a rational function in order to graph it. In particular, asymptotes can be used as a guide to sketch the graphs rational functions. We will learn to graph by looking at two examples.

Graphing rational functions where the degree of the numerator is equal to the degree of the denominator.

Consider the following rational function,

To determine what this function looks like, we must first write f (x) in lowest terms by canceling any common factor, which will allow us to find its asymptotes. To find the factors of Q(x), we set Q(x) = 0 and solve for x as,

Q(x) = x² − 2x = x(x − 2) = 0,

where x = 0 and x = 2 are the zeros of Q(x). To find the factors of P(x), we set P(x) = 0 and solve for x as,

P(x) = x²− x − 2 = (x + 1)(x − 2) = 0,

where x = −1 and x = 2 are the zeros of P(x). Therefore, using the factors of Q(x) and P(x), we can rewrite f (x) as,

Now that we have written f (x) in lowest terms, we see that the degree of the numerator equals the degree of the denominator, indicating that has horizontal asymptote y = a/b where a is the leading coefficient of P (x) and b is the leading coefficient of Q(x). Thus, the horizontal asymptote is y = 1; this means that f (x) approaches the constant function y = 1 as x → ± ∞. We also notice that f (x) has a vertical asymptote given by the equation x = 0. Lastly, x = 2 is a zero of both Q(x) and P(x), which means that x = 2 is not in the domain of f (x). Thus, there is a hole in the graph of f (x) at x = 2.

Therefore, we have found that

• f(x) approaches 1 as |x| gets large,
• x = 0 is a vertical asymptote of f(x),
• x = −1 is a zero of f(x) because x = −1 is a zero of P(x),
• there is a hole in the graph of f(x) at x = 2.

Since x = 0 is a vertical asymptote of f(x), we can determine if the function is positive or negative to the left and right of the asymptote. To do this, we can substitute a value of x a little to the left (say x = −0.01) and a little to the right of zero (say x = 0.01). We find that,

so f(x) is negative to left of x = 0 and positive to the right of x = 0. Using this information and the asymptotes as a guide, the graph of f(x) looks like:

Graphing rational functions where the degree of the numerator differs from the degree of the denominator.

Now consider the rational function,

First we find any asymptotes by canceling any common factors thus writing f (x) in lowest terms. To find the factors of Q(x), we set Q(x) = 0 and solve for x as,

Q(x) = 2x - 5 = 0,

we find that x = 5/2 is a zero of Q(x). Thus, (x - 5/2) is a factor of Q(x). Now we must check to see if x = 5/2 is a zero of P(x). We can do this simply by evaluating,

Because P(5/2) ≠ 0, x = 5/2 is not a zero of P(x), we know that f(x) is already in lowest terms. Since the degree of the numerator is exactly one greater than the degree of the denominator,
f(x) has an oblique asymptote. We find this asymptote using polynomial division as,

where the quotient y = x + 4 is the oblique asymptote. Therefore, f(x) approaches the line
y = x + 4 as x → ±∞. Furthermore, since x = 5/2 is a zero of Q(x) but not P(x), it is a vertical asymptote of f(x).

Therefore, we have found that

• f(x) approaches the line y = x + 4 as |x| gets large,
• x = 5 / 2 is a vertical asymptote of f(x),
• x = 0 and x = −3 / 2 are zeros of f(x).

Using this information and the asymptotes as a guide, the graph of f(x) looks like:

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Now try some problems that will test your understanding of rational functions.

Problems