You are correct. The rational function becomes undefined when the denominator equals zero.

Transcription

1 3.4. RATIONAL FUNCTIONS. A rational function is a function of the form where P() and Q() are polynomial functions and Q(). It is important be able identify the domain of rational functions. The domain is the set of all real numbers ecluding those values that make Q() =. What happens when Q() =? You are correct. The rational function becomes undefined when the denominar equals zero. In this lesson, we will focus on the behavior of the graphs of rational functions. We will eamine the graphs near the values that make these functions undefined. But first. A quick review. The following pics are useful in determining the graphical behavior of polynomial functions. The degree and the leading coefficient of a polynomial function. o What happens when the degree of a polynomial is odd? When it is even? o What happens when the leading coefficient is positive? When it is negative? Descartes' Rule of signs. o What happens when the number of sign variations for P() is odd? When it is even? o yintercept. What happens when the number of sign variations for P() is greater than or equal 2.

2 You will apply these pics and others determine the graphical behavior of rational functions. Let's look at the domain and the behavior of the graph of some rational functions. EXAMPLE :, the simplest rational function. The domain of f() is the set of all real numbers ecept. Now eamine the table of ordered pairs determine the behavior of f(). values from f() ( ) ( ), , values from., ( ) undefined ( )

3 , ( ) f() undefined, ( ) ( ) ( ) Note: As es zero ( ) from the left, f() decreases without bound As es zero ( ) from the right, f() increases without bound. Since cannot equal zero, a break occurs and the graph of f() will never intersect the vertical line =. This vertical line is called a vertical asympte. Also note: As decreases or increases in value, the f() es zero. We say that es zero as the absolute value of increases, (f() as ). The graph of f() gets closer and closer the horizontal line y =. This horizontal line is called a horizontal asympte. Unlike vertical asymptes, the graphs of some rational functions may cross the horizontal asympte. Let's eamine the graph.

4 The graph of f() is a hyperbola. EXAMPLE 2: The domain of f() is the set of all real numbers ecept 3. Now eamine the table of ordered pairs determine the behavior of f(). f() ( ) ( ) 3, values from values from , 3. 2, 3 ( 3) 3 undefined ( )

5 3 ( 3) f() undefined ( ) 2 ( ) 2, 2, ( ) Note: As es 3 ( 3) from the left, f() decreases without bound As es 3 ( 3) from the right, f() increases without bound. Since cannot equal 3, a break occurs and the graph of f() will never intersect the vertical line = 3. Also note: As decreases or increases in value, f() es zero. We say that f() es zero as the absolute value of increases (f() ). as The graph of f() gets closer and closer the horizontal line y =. Let's eamine the graph.

6 EXAMPLE 3: The domain of f() is the set of all real numbers ecept 2 and 3. values in each interval of the domain will be selected. Now eamine the table of ordered pairs determine the behavior of f(). f() ( ) values from ( ) 2 ( 2) ( ) values from undefined

7 f() undefined Note: 2 ( 2) ( ) values from 3 3 ( 3) ( 3) f() undefined ( ) As es 2 ( 2) from the left, f() increases without bound As es 2 ( 2) from the right, f() decreases without bound. As es 3 ( 3) from the left, f() decreases without bound As es 3 ( 3) from the right, f() increases without bound. undefined ( ) ( ) ( ) Since cannot equal 2 and cannot equal 3, two breaks occurs and the graph of f() will never intersect the vertical lines = 2 and = 3. Also note: As decreases or increases in value, f() es. We say that f() es as the absolute value of increases, (f() as ). The graph of f() gets closer and closer the horizontal line y =.

8 Let's eamine the graph. A definition for vertical and horizontal asympte of the graph of, written in lowest terms, can be stated as follows: The line = a is a vertical asympte if the absolute value of f() es infinity ( f() ) as es a ( a). The line y = b is a horizontal asympte if f() es b (f() absolute value of es infinity ( ) b) as the Now, let's find out how locate these asymptes. VERTICAL ASYMPTOTE The vertical asympte is obvious. Each ecluded value of the domain is a vertical asympte. To determine which values are ecluded, set the denominar equal zero and solve. HORIZONTAL ASYMPTOTE To locate the horizontal asympte is more interesting. Let f be a rational function

9 given by, written in lowest terms There are three possibilities based on the trichomy property ( m < n, m = n, or m > n ) for (m) equal the degree of the numerar and (n) equal the degree of the denominar. CASE ( m < n ): CASE 2 ( m = n ) : The degree of the numerar is less than the degree of the denominar. The ais is the horizontal asympte. The line y = is the horizontal asympte. The degree of the numerar is equal the degree of the denominar. The quotient of the leading coefficients is the horizontal asympte. CASE 3 ( m > n ): The line is a horizontal asympte. The degree of the numerar is greater than the degree of the denominar. The graph of f has no horizontal asympte. If the degree of the numerar is eactly one more than the degree of the denominar, the graph of f may have an oblique (slant neither vertical or horizontal) asympte. An oblique asympte can be determined by finding the quotient of the two polynomials disregarding any remainders.

10