Rational functions are functions that can be written in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomials. Rational functions play a crucial role in mathematics, physics, and engineering, among other fields.
One important characteristic of rational functions is the presence of vertical asymptotes. A vertical asymptote is a vertical line that the graph of the rational function approaches but never touches.
Read on to explore what vertical asymptotes are, why they are essential, and, most importantly, how to find them in simple and more complex rational functions. Check out this article to also explore how to find the horizontal asymptote.
What Are Vertical Asymptotes?
Vertical asymptotes are vertical lines x = a, at which a function is approaching positive (∞) or negative (- ∞) infinity as the inputs approach a. In other words, vertical asymptotes occur when the denominator of a rational function is equal to zero, and the numerator is not equal to zero (Source: Andrews University)
Rational functions may also have horizontal and oblique asymptotes ( or slant asymptote), points of discontinuity, and other interesting features. Analyzing such features can provide us with insights into the behavior and properties of the functions.
Why Vertical asymptotes are essential? Vertical asymptotes help to identify restrictions for rational functions, which is essential when calculating limits and finding relative extrema. Vertical asymptotes are critical features of rational functions that often appear in calculus and algebra courses.
However, for most students, finding them can be tricky, but with the right strategy, you can easily find vertical asymptotes of rational functions.
How to Find Vertical Asymptotes of Rational Functions
To find vertical asymptotes, you will need to identify the values of x that make the denominator (bottom) of your rational function equal to zero, which means that you will have to solve for x when the denominator equals zero.
Let’s consider the denominator of the rational function, q(x). A vertical asymptote occurs at a point x = a if and only if q(a) = 0 and p(a) is not equal to 0. Therefore, the vertical asymptotes of the rational function f(x) = p(x)/q(x) are the vertical lines x = a, where a is any real number that makes q(a) = 0 and p(a) is not equal to 0.
For example, the rational function f(x) = (x2 – 4)/(x2 – 9).
The denominator of this rational function, q(x) = x2 – 9, factors as (x – 3)(x + 3). Hence, the rational function has two vertical asymptotes, x = 3 and x = -3.
To verify these asymptotes, we need to check whether p(x) is not equal to 0 when x = 3 or x = -3. We can see that p(3) = -1 and p(-3) = 1, which confirms that x = 3 and x = -3 are vertical asymptotes of the function.
Some simple functions have only one vertical asymptote, while others may have two or three. An essential thing to remember is that vertical asymptotes never cross the x-axis, and they are always vertical lines. Also, vertical asymptotes help to identify the domain and range of the function as they limit where the function can go and to which values of y it is restricted.
Check out this video to find the domain, range, and asymptotes of rational functions using the Texas Instruments ti-84 plus CE.
What to read next:
- How to Easily do Long Division with Polynomials (Including a helpful steps by step guide)
- Elimination Method: Definition and how to do it.
- Slope Intercept Form: What it is, how to find, and its applications.
Wrapping Up
Vertical asymptotes are essential features of rational functions that help to identify restrictions, domains, and ranges. To find the vertical asymptotes of a rational function, we need to consider the denominator of the function and identify the values of x that make the denominator equal to 0.
Remember that vertical asymptotes occur at values of x that make the denominator equal to 0 and the numerator nonzero.