Section 4.6Rational Functions and Their Graphs

Section 4.6Rational Functions and Their Graphs

Section 4.6Rational Functions and Their Graphs

Objectives

  1. Finding the Domain and Intercepts of Rational Functions
  2. Identifying Vertical Asymptotes
  3. Identifying Horizontal Asymptotes
  4. Using Transformations to Sketch the Graphs of Rational Functions

7. Sketching Rational Functions

In this section, we will investigate the properties and graphs of rational functions.

Definition Rational Function

A rational function is a function of the form where g and h are polynomial functions such

Objective 1: Finding the Domain and Intercepts of Rational Functions

Since polynomial functions are defined for all values of x, it follows that rational functions are defined for all values of x except those for which the denominator is equal to zero. If has a y-intercept, it can be found by evaluating provided that is defined. If has any x-intercepts, they can be found by solving the equation(provided that g and h do not share a common factor).

Objective 2: Identifying Vertical Asymptotes

To begin our discussion of the graphs of rational functions, we will first look at a basic function that was introduced in Section 3.3, the reciprocal function, . The function is defined everywhere except at; therefore, the domain of is . You can see from the graph of sketched below that as the values of x get closer to 0 from the right side of 0, the values of increase without bound. Mathematically, we say, “as x approaches zero from the right, approaches infinity.” In symbols this is written:

The “+” symbol indicates that we are only looking at values of x that are located to the right of zero. Similarly, as x gets closer to 0 from the left side of 0, the values of approach negative infinity. Symbolically we can write the following:

The “” symbol indicates that we are only looking at values of x that are located to the left of zero.

The graph of

The graph of the function gets closer and closer to the line (the y-axis) as x gets closer and closer to 0 from either side of 0, but the graph never touches the line . The line is said to be a vertical asymptote of the graph of . Many rational functions have vertical asymptotes. The vertical asymptotes occur when the graph of a function approaches positive or negative infinity as x approaches some finite number a as stated in the following definition.

DefinitionVertical Asymptote

The vertical line is a vertical asymptote of a function if at least one of the following occurs:

A rational function of the form where and have no common factors will have a vertical asymptote at if .

It is essential to cancel any common factors before locating the vertical asymptotes.

The situation that arises when and share a common factor will be discussed in Objective 5 of this section.

If there is an x-intercept near the vertical asymptote, it is essential to choose a test value that is between the x-intercept and the vertical asymptote.

It is absolutely crucial to choose test values that are strictly between the x-intercept and the vertical asymptote.

Objective 3: Identifying Horizontal Asymptotes

The graph of is sketched again below. The graph shows that as the values of x increase without bound (as), the values of approach 0 (). Similarly, as the values of x decrease without bound (as), the values of approach 0 (). In this case, the line is called a horizontal asymptote of the graph of .

The line is a horizontal

asymptote of the graph of

Definition Horizontal Asymptote

A horizontal line is a horizontal asymptote of a function if the values of approach some fixed number as the values of approach or .

The line is a horizontal asymptote The line is a horizontal asymptote The line is a horizontal asymptote

because the values ofapproach because the values ofapproach because the values ofapproach

as x approaches . as x approaches . as x approaches .

Properties of Horizontal Asymptotes of Rational Functions

  • Although a rational function can have many vertical asymptotes, it can have at most one horizontal asymptote.
  • The graph of a rational function will never intersect a vertical asymptote but may intersect a horizontal asymptote.
  • A rational function that is written in lowest terms (all common factors of the numerator and denominator have been cancelled) will have a horizontal asymptote whenever the degree of is greater than or equal to the degree of .

The third property listed above tells us that a rational function (written in lowest terms) will have a horizontal asymptote whenever the degree of the denominator is greater than or equal to the degree of the numerator. If the degree of the denominator is less than the degree of the numerator, then the rational function will not have a horizontal asymptote. We now summarize a technique for finding the horizontal asymptotes of a rational function.

Finding Horizontal Asymptotes of a Rational Function

Let ,

whereis written in lowest terms, is the degree of , and m is the degree of .

  • If , then is the horizontal asymptote.
  • If , then the horizontal asymptote is , the ratio of the leading coefficients.
  • If , then there are no horizontal asymptotes.

Objective 4: Using Transformations to Sketch the Graphs of Rational Functions

We now introduce another basic rational function, . Like the reciprocal function, the domain of is . The denominator, , is always greater than zero, and this implies that the range includes all values of y greater than zero. The graph has a y-axis vertical asymptote () and an x-axis horizontal asymptote (). Below are the graphs of and along with some important properties of each. Knowing these two basic graphs will help us sketch more complicated rational functions.

The graphs of and

Properties of the graph of Properties of the graph of

Domain: Domain:

Range: Range:

No interceptsNo intercepts

Vertical Asymptote: Vertical Asymptote:

Horizontal Asymptote: Horizontal Asymptote:

Odd function Even function

The graph is symmetric about the origin.The graph is symmetric about the y-axis.

Objective 7:Sketching Rational Functions

Steps For Graphing Rational Functions of the Form

1.Find the domain.

2.If and have common factors, cancel all common factors determining the x-coordinates of any removable discontinuities and rewrite f in lowest terms.

3. Check for symmetry.

If , then the graph of is odd and thus symmetric about the origin.If , then the graph of is even and thus symmetric about the y-axis.

4.Find the y-intercept by evaluating.

5. Find the x-intercepts by finding the zeros of the numerator of f, being careful to use the new numerator if a common factor has been removed.

6. Find the vertical asymptotes by finding the zeros of the denominator of f, being careful to use the new denominator if a common factor has been removed. Use test values to determine the behavior

of the graph on each side of the vertical asymptotes.

7. Determine if the graph has any horizontal asymptotes.

8. Plot points, choosing values of x between each intercept and choosing values of x on either side of the all vertical asymptotes.

9. Complete the sketch.