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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Bob Bruner*

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

A rational function arises from the ratio of two polynomial expressions. The graphs of rational functions often have distinct characteristics. In this lesson, we look at how to analyze some of those characteristics.

**Rational functions** are defined as the ratio of two polynomial expressions. For example, suppose we're given two simple linear polynomial functions:

f1 = 10*x* + 6

f2 = *x* - 1

We can compose a rational function by simply taking their ratio.

f1 / f2 = (10*x* + 6) / (*x* - 1)

The graph of the resulting function is surprisingly complex for such simple inputs. We can see that this derived function consists of two distinct parts. Depending on the polynomial expressions found in the numerator and denominator, graphs of rational functions may take on various complex shapes. These shapes are defined in part by the effective domain and range of the function.

The **domain** of a function consists of all the allowable values for the independent variable *x*. Similarly, the **range** of the function consists of all the possible values for the dependent variable *y*. If we look more closely at our graph, we see that it approaches, but never actually arrives at, certain values in *x* and *y*. In this case, both portions of the graph function are asymptotic to those values. Any asymptote that crosses the *x*-axis parallel to the *y*-axis can be referred to as a **vertical asymptote** of the function. Similarly, any asymptote that cross the *y*-axis parallel to the *x*-axis is referred to as a **horizontal asymptote**.

Here is our graph with the vertical and horizontal asymptotes plotted as dashed lines:

The vertical and horizontal asymptotes help us to find the domain and range of the function. We see that the vertical asymptote has a value of *x* = 1. From this, we can state that the domain of this function consists of all values in *x*, except for 1. If we look back at our original function, note that the denominator is the term *x* - 1. Using 1 as a value of *x* would result in a value of 0 for the denominator. Because division by 0 is undefined, it's mathematically reasonable that the domain of our rational function consists of all values except for 1. In the case of more complex polynomial equations, we can factor the denominator and then solve for 0 in each of the individual terms. Any of those values will be outside the domain of the function, as the function is undefined at those locations.

In a similar manner, we can graphically see that a horizontal asymptote occurs at the value of *y* = 10. From this, we can state that the range of this function consists of all values in *y* except for 10.

There are a few rules we can use to help arrive at these locations. These rules are based on the value of the largest exponent found in the numerator and denominator of the rational function. Consider the general form of a rational function, where *m* is the largest exponent in the numerator and *n* is the largest exponent in the denominator. If *m* less than *n*, then the *x*-axis itself will be the horizontal asymptote. In this case, the denominator will always have a larger magnitude than the numerator and will approach, but never arrive at, a value of *y* = 0 (i.e., the *x*-axis). If *m* = *n*, then the line *y* = *a*/*b* is the horizontal asymptote. This is the case in our simple function: *m* and *n* both have a value of 1 and *y* = *a*/*b* = 10/1 = 10. If *m* = *n* + 1 then there will be no horizontal asymptote. In this case, we will have what's called a slant asymptote, which we will explore in more detail soon. Any horizontal asymptote will define a location that is not part of the range of the rational function. Rather than deriving these values using rules and calculations, these locations are often easiest to find simply by looking at the graph of the rational function.

Not all asymptotes found in the graphs of rational functions are parallel to the *x*- or *y*-axis. They can occur at any angle. These asymptotes are referred to as **slant asymptotes**, or sometimes **oblique asymptotes**. In fact, slant asymptotes arise in the last case mentioned for finding horizontal asymptotes. This is the case where the largest exponent found in the numerator is one degree larger than the largest exponent found in the denominator. For example, consider the rational function: f(*x*) = (*x*2 - 8*x* - 11) / (*x* + 4). The graph of the function appears as follows:

Because the denominator is *x* + 4, we expect a vertical asymptote at *x* = -4, but because the largest exponent in the numerator is larger than the largest exponent in the denominator by 1 degree, we also have a slant asymptote. The line that this asymptote follows is a simple calculation. We only need to divide the numerator by the denominator, ignoring any remainder terms that would not affect the resulting linear equation. In this example, the result is *y* = *x* - 12.

**Rational functions** are defined as the ratio of two polynomial expressions. When graphed, these functions often have unique shapes that are controlled, in part, by the function's domain and range. For a rational function defined as *y* = f(*x*)1 / f(*x*)2, the **domain** consists of all possible values in *x* and the **range** consists of all possible values in *y*. **Vertical asymptotes** occur wherever the rational function would result in division by zero and defined points outside the domain. **Horizontal asymptotes** occur wherever the rational function approaches, but never arrives at, a specific value in *y* and are controlled by the relative degree of the largest exponential factors in the numerator and denominator of the rational function. If the largest exponential factor of the numerator is greater than the one in the denominator by one degree, then **slant asymptotes** will occur on the graph.

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- How to Multiply and Divide Rational Expressions 8:07
- Multiplying and Dividing Rational Expressions: Practice Problems 4:40
- How to Add and Subtract Rational Expressions 8:02
- Practice Adding and Subtracting Rational Expressions 9:12
- How to Solve a Rational Equation 7:58
- Horizontal and Vertical Asymptotes 7:47
- Graphing Rational Functions That Have Linear Polynomials: Steps & Examples 7:55
- Graphing Rational Functions That Have Polynomials of Various Degrees: Steps & Examples 8:59
- Analyzing the Graph of a Rational Function: Asymptotes, Domain, and Range 5:50
- Go to Rational Expressions and Functions

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