RATIONAL FUNCTIONS. Finding Asymptotes..347 The Domain Finding Intercepts Graphing Rational Functions

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1 RATIONAL FUNCTIONS Finding Asymptotes..347 The Domain Finding Intercepts Graphing Rational Functions

2 Objectives The ollowing is a list o objectives or this section o the workbook. By the time the student is inished with this section o the workbook, he/she should be able to Find the vertical asymptotes o a rational unction. Determine i the unction has a horizontal or oblique asymptote. Find the horizontal asymptote o a rational unction i it eists. Find the oblique asymptote o a rational unction i it eists. Find the domain o a rational unction. Find the intercepts o a rational unction. Find the y intercept o a rational unction. Graph a rational unction. Math Standards Addressed The ollowing state standards are addressed in this section o the workbook. Algebra II 3.0 Students are adept at operations on polynomials, including long division. 4.0 Students actor polynomials representing the dierence o squares, perect square trinomials, and the sum and dierence o two cubes. 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational epressions with monomial and polynomial denominators and simpliy complicated rational epressions, including those with negative eponents in the denominator. 8.0 Students solve and graph quadratic equations by actoring, completing the square, or using the quadratic ormula. Students apply these techniques in solving word problems. They also solve quadratic equations in the comple number system. 5.0 Students determine whether a speciic algebraic statement involving rational epressions, radical epressions, or logarithmic or eponential unctions is sometimes true, always true, or never true. Mathematical Analysis 6.0 Students ind the roots and poles o a rational unction and can graph the unction and locate its asymptotes. 346

3 Finding Asymptotes Rational unctions have various asymptotes. The ollowing will aid in inding all asymptotes o a rational unction. The irst step to working with rational unctions is to completely actor the polynomials. Once in actored orm, ind all zeros. Vertical Asymptotes The Vertical Asymptotes o a rational unction are ound using the zeros o the denominator. For Horizontal Asymptotes use the ollowing guidelines. I the degree o the numerator is greater than the degree o the denominator by more than one, the graph has no horizontal asymptote.(none) I the degree o the numerator is equal to the degree o the denominator, the horizontal asymptote is the ratio o the two leading coeicients.(y #) I the degree o the numerator is less than the degree o the denominator, the horizontal asymptote is zero. (y 0) Oblique Asymptotes I the degree o the numerator is greater than the degree o the denominator by one, there is an oblique asymptote. The asymptote is the quotient numerator divided by the denominator. An asymptote is like an imaginary line that cannot be crossed. All rational unctions have vertical asymptotes. A rational unction may also have either a horizontal or oblique asymptote. A rational unction will never have both a horizontal and oblique asymptote. It is either one or the other. Horizontal asymptotes are the only asymptotes that may be crossed. The vertical asymptotes come rom zeroes o the denominator. ( ) ( + )( 3) and 3 Here is a rational unction in completely actored orm. The zeros o the denominator are - and 3. Thereore, these are the vertical asymptotes o the unction. Since an value o - or 3 would create a zero in the denominator, the unction would be undeined at that location. As a result, these are the vertical asymptotes or this unction. 347

4 In this same unction, the degree o the numerator is less than the degree o the denominator, thereore, the horizontal asymptote is y 0. When inding the oblique asymptote, ind the quotient o the numerator and denominator. I there are any remainders, disregard them. You only need the quotient. The graph o the unction can have a either a horizontal asymptote, or an oblique asymptote. You can not have one o each. This particular unction does not have an oblique asymptote. Here is an eample with an oblique asymptote. Find the oblique asymptote o the rational unction ( ) Dividing the polynomials, the quotient +9 is ound. y + 9 This is the equation or the oblique asymptote o the unction. Notice the remainder o the division problem is disregarded. It plays no part in the equation or the oblique asymptote. Finally, let us look at a rational unction where the degree o the numerator is equal to the degree o the denominator Find the horizontal asymptote or the rational unction ( ). 3 7 ( ) y 3 Notice the degree o the numerator is the same as the degree o the denominator. Since the degree o the numerator equals that o the denominator, the equation or the horizontal asymptote is the ratio o the two leading coeicients. 348

5 Find all asymptotes o the ollowing unctions. (Do not graph these) 7 3 A) ( ) B) + ( ) 5 C) ( ) 5 D) ( ) E) ( ) F) ( ) G) ( ) 3 H) ( ) I) ( )

6 The Domain The domain o a rational unction is ound using only the vertical asymptotes. As previously noted, rational unctions are undeined at vertical asymptotes. The rational unction will be deined at all other values o the domain. ( ) ( + )( 3) and 3 Here is a rational unction in completely actored orm. Since the zeros o the denominator are - and 3, these are the vertical asymptotes o the unction. Thereore, the domain o this unction is (, ) (,3) ( 3, ). Notice there are two vertical asymptotes, and the domain is split into three parts. This pattern will repeat. I there are 4 vertical asymptotes, the domain o that unction will be split into 5 parts. Find the domain o each o the ollowing rational unctions. A) ( ) 7 3 B) + ( ) 5 4 C) ( ) 5 D) ( ) E) ( ) 3 F) ( ) G) ( ) 3 H) ( ) I) ( )

7 Finding Intercepts We have ound that the zeros o the denominator o a rational unction are the vertical asymptotes o the unction. The zeros o the numerator on the other hand, are the intercepts o the unction. 9 Find all and y intercepts o the unction ( ). ( + 3)( 3) ( ) Write out the unction in completely actored orm. Now, ind the zeros o the numerator 3 and 3 Look at the original unction. 9 ( ) y 9 The intercepts are ( 3,0) and ( 3,0) The y intercept is ( 0,9 ) These are the intercepts o the unction. From here, substitute zero or, and ind the y intercept, which in this case will be the ratio o the two constants. This is the y intercept o the unction. In this case, it is the ratio o the two remaining constants once zero is substituted in or. I there is no constant in the denominator, then there will be no y intercept as 0 is a vertical asymptote and the graph is undeined at the y ais. As demonstrated above, the y intercept o a rational unction is the ratio o the two constants. Like always, substitute zero or, and solve or y to ind the y intercept. Find the and y intercepts o each rational unction. A) ( ) 7 3 B) + ( ) 5 4 C) ( ) 5 D) ( ) E) ( ) 3 F) ( )

8 Graphing Rational Functions We really have no standard orm o a rational unction to look at, so we will concentrate on the parent unction o ( ). The ollowing pages illustrate the eects o the denominator, as well as the behavior o ( ). A graphing calculator may be used to help get the overall shape o these unctions. DO NOT, however, just copy the picture the calculator gives you. ( ) Here, the vertical asymptote is at 0, and the horizontal asymptote is y0. ( ) + ( ) The graph o this unction shits let. The graph o this unction shits right. The range or each o these unctions is (, 0) ( 0, ). There is no way to tell what the range o a rational unction will be until it is graphedt. Remember, the curve may cross the horizontal ais. 35

9 ( ) Here, the vertical asymptote is at 0, and the horizontal asymptote is y0. ( ) ( ) The graph o this unction is a relection o the parent unction. ( ) 3 Notice how aects the unction. Normally, one side o the unction would go up, and the other would go down. Since there is no way to get a negative number in the denominator, both sides are going in the same direction. The graph o the unction to the let lips upside down, similar to ( ), and shits right 3. What happens here is a - is actored out o the denominator, changing the unction to the ollowing. ( ) ( ) As a result, this graph is a combination o shiting the graph and relecting it about the horizontal asymptote. 353

10 Eample 3 Graph the unction ( ) and domain. 3 ( ) ( + 3)( 4). Be sure to ind all asymptotes, and y intercepts, and the range The irst step is to completely actor the rational unction. Vertical Asymptotes are 3 and 4 Looking at the original unction, the horizontal asymptote is y 0. The intercept is ( 3,0 ) The zeros o the denominator are the vertical asymptotes o the unction. I the degree o the numerator is less than the degree o the denominator, the horizontal asymptote is y 0. The zero o the numerator is the intercept o the unction. The y intercept is 0, 4. Substituting zero or and evaluating the ratio o the two constants, -3 and -. Yields a y intercept o 4. The domain o the unction is, 3 3, 4 4, ( ) ( ) ( ) The domain is ound using the vertical asymptotes. The domain here is all real numbers ecept -3 and 4. The range should not be ound without irst graphing the unction. ( ) 3 Due to limitations in the graphing sotware, the graph to the let is incomplete. Each o these lines is continuous. Beore attempting to graph the unctions, graph all asymptotes using broken lines. This will ensure that a vertical asymptote is not crossed. To get the general shape o the equation, use a combination o the and y intercepts that were ound, and plug in values or close to the vertical asymptote. Based on the properties o all rational unctions, it should be obvious how these curves behave on the outer intervals o these unctions. They will always ride along the asymptotes in these areas. V.A.: 3, 4 H.A.: y 0 -int: ( 3,0 ) y-int: 0, 4 Range: (, ) Domain: (, 3) ( 3, 4) ( 4, ) Here is a list o all required inormation needed or each rational unction. Since the graph o the unction crossed the horizontal asymptote in the interval ( 3, 4), the range o this unction is all real numbers. These procedures must be used when graphing any rational unction. 354

11 Match the appropriate graph with its equation below. Eplain why each o the solutions is true. A B C D E F ) ( ) ) ( ) 3 3) ( ) 4) ( ) + 5) ( ) 6) ( ) + 355

12 A graphing calculator may be used to help get a picture o the curve that will be created, but simply copying the picture shown in the calculator is unwise. What is the problem with the picture o rational unctions in graphing calculators? Sketch the graph o each o the ollowing unctions. Be sure to ind all asymptotes, and y intercepts, and the range and domain o each o the ollowing. A) ( ) 4 356

13 B) ( ) + C) ( )

14 D) ( ) E) ( ) + 358

15 + F) ( ) 9 G) ( ) + 359

16 H) ( ) 4 I) ( )

17 J) ( ) + 6 K) ( ) 4 36

18 L) ( ) + + M) ( )

19 N) ( ) ( ) O) ( ) + 363

20 Checking Progress You have now completed the Rational Functions section o the workbook. The ollowing is a checklist so that you may check your progress. Check o each o the objectives you have accomplished. The student should be able to Find the vertical asymptotes o a rational unction. Determine i a rational unction has a horizontal or oblique asymptote. Find the horizontal asymptote o a rational unction i it eists. Find the oblique asymptote o a rational unction i it eists. Find the domain o a rational unction. Find the intercepts o a rational unction. Find the y intercept o a rational unction. Graph a rational unction. 364