9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational unction that we have seen beore is ( ). As we can see, rational unctions have limitations on their domains. The denominator can not be zero. So what happens when the denominator is zero? Lets investigate.. Lets see what happens as we get closer and closer to zero rom the positive and negative sides o zero. Here is a table o values. We will consider our unction ( ) () 0. 0 0.0 00 0.00 000 0.000 0,000 () - - -0. -0-0.0-00 -0.00-000 -0.000-0,000 As we can see, the unctional values continue to increase as we get closer to zero rom the positive side and they continue to become large negative rom the negative side. We say that as tends to zero, () tends to ininity or negative ininity, respectively. This gives us the graph o When this situation happens (on one side or the other, or both) we have what is called a vertical asymptote. Property or Vertical Asymptotes A rational unction, which has no common actors in the numerator and denominator, has vertical asymptotes at all points where the denominator is zero. So in order to ind the vertical asymptotes we simply need to make sure there are no common actors and then ind when the denominator is zero. These values would be the vertical asymptotes.
is that as the values o increase, the values o () get closer to zero. And as the values become increasingly large negatives the values o () get closer to zero as well. The net thing we notice about our unction ( ) This idea is one o a horizontal asymptote. A horizontal asymptote is any value that a unction gets close to as the values o get increasingly large in the positive or negative direction. We use the ollowing property to determine the horizontal asymptote. The proo o this property is reserved or precalculus. Property or Horizontal Asymptotes P Let ( ) ( ) n m where the leading term o P is a and the leading term o Q is b. Then Q( ) a. I n < m, then the horizontal asymptote is at y 0. b. I n > m, then the unction has no horizontal asymptote. a c. I n m, then the horizontal asymptote is at y. b So we need only to compare the leading terms o the numerator and denominator to determine the horizontal asymptotes. Eample : Determine all the asymptotes o ( ) 5. 5 First we need to actor the numerator and denominator to see i we can cancel any common actors. ( ) 5 5 ( )( ( )( ) ) Now we will start with the vertical asymptotes. The vertical asymptotes occur when the denominator is zero. So set the denominator equal to zero and solve. We get 5 0 ( )( ) 0 So the vertical asymptotes are at 0 0 and.
Now the horizontal asymptotes are ound by using the property above. Since the degree o the numerator and denominator are the same, the horizontal asymptote occurs at the ratio o the leading coeicients. We get y. Net we need to discuss inding the intercepts o a rational unction. Recall, we ind the y-intercept o an equation by letting 0 and solve or y. We do the same thing in a rational unction. Also, recall, we ind the -intercepts o an equation by letting y0 and solve or. In a rational unction however, this is the same as setting the numerator equal to zero and solving. The reason or this is that a raction can only be zero when its numerator is zero. Eample : Find the intercepts o ( ) 5 5. We will start with the y-intercept. So we let 0 and get y ( 0) 5( 0) ( 0) 5( 0) For the -intercepts we set the numerator equal to zero and solve. 5 0 ( )( ) 0 0 0 The last thing we need in order to graph a rational unction is a sign chart. A sign chart is a chart used to ind where an equation is entirely positive and negative. A sign chart tells us i the graph o a unction is above the -ais (positive) or below the -ais (negative). Clearly, there are only a ew ways in which a unction can change rom positive to negative. One way is to actually cross the -ais, becoming an intercept, and the other way is i there is an asymptote. Once we have a sign chart we can use this inormation with our intercepts and asymptotes to generate the graph o a rational unction. Here are the steps involved in generating a sign chart.
Creating a Sign Chart or a Rational unction. Find and plot on a number line, all -intercepts and vertical asymptotes. Use an open circle or the vertical asymptotes and a closed circle or the -intercepts.. Use the number line rom step to generate your intervals to be tested. We generally go rom one asymptote or intercept to another to generate these intervals.. Test any value in the intervals rom step. The sign o this value will characterize the sign o the entire interval.. Complete the sign chart by placing the sign o each interval directly above the interval on the graphed number line. Eample : Construct the sign chart or ( ) 5 5. From eamples and we know the vertical asymptotes and -intercepts or ( ) 5 5. So we can put them on a number line as ollows. O - Note: We need to use open circle on the asymptotes since those values are not in the domain and thus are not deined, and illed circles on the -intercepts since that is the spots at which the graph would actually be touching the -ais. So now we go rom asymptote or intercept to asymptote to intercept to create our intervals. This would give us the ollowing intervals: ( ), (, ), (,), (,), (, ), We now test a value out o each interval. Since we are only concerned about the sign o the unction in that interval, we only need to ind i the test value makes the interval positive or negative. For the irst interval (,) So the irst interval is positive. For the net interval (, lets test. ( ), ) we can test 0. ( ) 5( ) ( ) 5( ) 8 O
In a similar ashion we ind that (, ( ) Summarizing this on the sign chart we get ( 0) 5( 0) ( 0) 5( 0), ) is, ( ) is and (, ) O - - O - is. Now all we have to do is take all o the inormation we have ound and put it all on a graph. Eample : Graph ( ) 5 5. From eample we see that we have vertical asymptotes o, and horizontal asymptote o y. We start by graphing these on the rectangular grid with dashed lines. Then we plot the intercepts o (,0), (, 0 ) and (, ) around the asymptotes to inish the graph. 0. Finally we use the sign chart and the behavior Plotting points can see the act that the graph dips below the horizontal asymptote beore getting close to it. Lets now put all o this together in our last eample to graph a rational unction. Eample 5:
Graph ( ). First we should put the unction in actored orm. ( ) ( )( ) From this we can see that the -intercept is and the y-intercept is y. The vertical asymptotes occur at and. Also, since the numerator has a smaller degree than the denominator, we have that the horizontal asymptote is y 0. Now using the -intercepts and vertical asymptotes we can construct our sign chart. O - So our intervals to test are (, ), (, ), (, ), (, ). Testing a point in each interval will give us the ollowing sign chart So putting it all together we get the graph. O - O - O - 9.8 Eercises Find all vertical and horizontal asymptotes o the ollowing rational unctions.. ( ). ( ). ( ) 5. ( ) 5. ( ). ( ) ( )( ) 7 8 5 7. ( ) 8. ( ) 9. ( ) ( 5) 5
5 0. ( ). ( ) 5. ( ). ( ) 5 8. ( ) 5. ( ) Find all - and y-intercepts o the ollowing rational unctions.. ( ) 7. ( ) 8. ( ) 5 9. ( ) 0. ( ). ( ) ( )( ) ( ) 5. ( ) ( 7)( ). ( ). ( ) 8 5 7 5 5. ( ). ( ) 0 9 9 00 7. ( ) 8. ( ) 5 0 8 9. ( ) 0. ( ) Graph the ollowing rational unctions. Find all asymptotes and intercepts.. ( ). ( ). ( ). ( ) 5. ( ). ( ) 7. ( ) 8. ( ) 9. ( ) ( )( ) 0. ( ). ( ). ( ) ( )( ) ( )( ) ( )( )
. ( ) ( )( ( )( ) ). ( ) ( ) ( )( ) 5. ( ) ( ). ( ) ( ) 7. ( ) 8. ( ) 9. ( ) 50. ( ) 5. ( ) 8 5. ( ) 8 5. ( ) 5. ( ) 55. ( ) 5. ( ) 57. ( ) 8 58. ( ) 5 8 59. ( ) 0. ( ) In the section we mentioned that the vertical asymptotes occurred where the denominator is zero, as long as the numerator and denominator have no common actors. In the case where they do have common actor it is a little dierent. For eample ( ) ( )( ) clearly have a common actor. I we cancel it out we end up with ( ),. Since the domain is still all real numbers ecept the graph o is just the line y- but with a hole at since the unction is not deined there. Use this idea to graph the ollowing unctions.. ( ). ( ). ( ). ( ) 5. ( ). ( ) 5