2.6 Rational Functions and Their Graphs

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1 .6 Ratioal Fuctios ad Their Graphs Sectio.6 Notes Page Ratioal Fuctio: a fuctio with a variable i the deoiator. To fid the y-itercept for a ratioal fuctio, put i a zero for. To fid the -itercept for a ratioal fuctio, set the uerator equal to zero EXAMPLE: Fid the ad y itercepts for y (0) First we will fid the y-itercept. We will put i a zero for. You will get: y 6. So our 0 poit is (0, -6). Net we will fid the -itercept. To do this we ust set the uerator (top) equal to zero. You will get: 0. Solvig this you will get = 6. So our poit is (6, 0). Asyptote: describes the behavior of a graph as or y approaches ifiity. There are two types of asyptotes. There is a vertical ad horizotal asyptote as show i the picture below. The vertical asyptote has a equatio that starts with = sice this is a vertical lie. The horizotal asyptote has a equatio y = sice this is a horizotal lie. Now we will show how you fid these algebraically. To fid the vertical asyptote: set the deoiator equal to zero ad solve for. Suppose we wated to fid the vertical asyptote of our previous eaple, y. To do this we eed to set the deoiator equal to zero, so we will have + = 0. The equatio will be = -. You would write your aswer i this for. You do t eed parethesis. That is oly for itercepts. To fid the horizotal asyptote: First we eed to defie soe variables. Let s look at the geeral for fro a ratio epressio: a a... f ( ) b b... Let be the highest power (degree) of the uerator. Let be the highest power (degree) of the deoiator. Let a be the uber that coes i frot of the with the highest power i the uerator. Let a be the uber that coes i frot of the s with the highest power i the deoiator.

2 I order to deterie the horizotal asyptote we eed to look at the ad. Sectio.6 Notes Page.) If < the the equatio of the horizotal asyptote is autoatically y = 0. a.) If = the the equatio of the horizotal asyptote is y. b 3.) If > the there is o horizotal asyptote. There is a oblique asyptote: You fid the oblique asyptotes by usig log divisio. More o this later. I our origial equatio y, if we were asked to fid the equatio of the horizotal asyptote we eed to see which rule applies. Here the highest power o top is the sae as the highest power o the botto, so rule applies. We kow a ad, so the horizotal asyptote is b y or just y =. EXAMPLE: Fid the itercepts ad asyptotes but DO NOT GRAPH: y 5 6 ( )( ) First you wat to factor to see if it ca be siplified further: y. This ca t be siplified ( )( 3) ayore, so ow let s fid the y-itercept. To do this we eed to put i a 0 for. Whe you do that you will ( 0)( 0) get y. So the y-itercept is 0,. To fid the -itercept we eed to set the top equal to (0 )(0 3) 6 6 zero. So we have ( )( ) 0. Solvig this we get, so our -itercepts are,0. Now let s fid the asyptotes. To fid the vertical asyptote we eed to set the botto equal to zero. So we have ( )( 3) 0 so = ad = 3. We will leave our aswer i this for. To fid the horizotal asyptote let s look at our origial equatio. The highest power o the top is the sae as the highest power o the botto, so we use rule agai. Our a ad b, so the horizotal asyptote is y so y = -. 3 EXAMPLE: Fid the itercepts ad asyptotes but DO NOT GRAPH: y 3 4 ( 3) 3 First you wat to factor to see if it ca be siplified further: y. You always wat ( ) ( ) to siplify if possible. So ow let s fid the y-itercept. To do this we eed to put i a 0 for. Whe you do you will get a zero i the deoiator, which is udefied. So there is o y-itercept. To fid the -itercept we eed to set the top equal to zero. So we have 3 0. Solvig this we get = 3, so this is (3, 0). Now let s fid the asyptotes. To fid the vertical asyptote we eed to set the botto equal to zero. So we have ( ) 0 so 0 ad We will leave our aswer i this for. To fid the horizotal asyptote let s look at our origial equatio. The highest power o the top is less tha the highest power o the botto, so we use rule. This says that the horizotal asyptote is autoatically y = 0.

3 EXAMPLE: Fid the asyptotes but DO NOT GRAPH: y Sectio.6 Notes Page 3 This does ot ask us to fid itercepts, so we will just fid the asyptotes. To fid the vertical asyptote we eed to set the botto equal to zero. So we have 0 so. We will leave our aswer i this for. To fid the horizotal asyptote let s look at our origial equatio. The highest power o the top is ore tha the highest power o the botto, so ow we have rule 3. This tells us there is o horizotal asyptote, but there is a oblique asyptote. We eed to fid this. I order to do that we ust use log divisio like we did i a previous sectio Fro doig the log divisio we get This is the equatio of the oblique 3 6 asyptote. We always igore the reaider. We just write y = Now we will look at soe special graphs. These are: y y y ad y. The oral graphs look like: We ca still graph by usig trasforatios like we did i previous sectios. EXAMPLE: Graph y usig trasforatios. 3 The 3 says we eed to ove the graph of y three places to the right. We will get

4 EXAMPLE: Graph y 6 9 usig trasforatios. Sectio.6 Notes Page 4 First we will factor: y ( 3)( 3) which ca be writte as y. The + 3 tells us that we eed to ( 3) ove y three places to the left. The egative tells us we eed to flip the graph horizotally. EXAMPLE: Graph y usig trasforatios. First I will divide everythig by to get y or y This tells us to ove the graph of y up oe uit ad the flip it horizotally. Now we will look at the graphs of ratioal fuctios. EXAMPLE: Fid the itercepts, asyptotes, ad graph of y. 9 First we will fid the -itercept by settig the top equal to zero: + = 0 so = -. We write (-, 0). To fid the y-itercept, put i a zero for : 0 y. You will get 0 9 y ad we write 0,. 9 9 To fid the vertical asyptote, set the botto equal to zero: 9 0 We get 3. For the horizotal asyptote, we otice the highest power o the top is less tha the highest power o the botto. Fro the previous sectio we kow that the horizotal asyptote is autoatically y = 0.

Sectio.6 Notes Page 5 Now we are ready for the graph. First plot the itercepts. The use dotted lies to draw i the asyptotes. The y-ais is oe asyptote, so we have a horizotal dotted lie at y = 0.

5 Sectio.6 Notes Page 5 Now we are ready for the graph. First plot the itercepts. The use dotted lies to draw i the asyptotes. The y-ais is oe asyptote, so we have a horizotal dotted lie at y = 0. The other asyptotes are 3. These will be vertical lies goig through 3 ad -3 o the -ais. Now we eed to graph. O the left ad right sides of the graph we have two possible ways the graph ca be draw as the picture below shows. Notice that o the eds the graph will follow oe asyptote, tur ad follow the other asyptote. This is always the case with ratioal epressios: We eed to chose whether the graph is above or below the -ais o each ed. I order to do this we eed to choose a test poits. We eed to choose a poit that is less tha egative 3 sice we wat to kow what the graph is doig to the left of vertical asyptote, = -3. I will choose -4. We will put our test poit ito the 4 3 origial equatio: y. We get y. Sice this uber is egative I kow that the graph ust ( 4) 9 7 be below the -ais. Now let s test a poit to the right of the vertical asyptote, = 3. I will choose 4 sice this is greater tha 3. Oce 4 agai we will put 4 i for i the origial equatio: y (4) 9 5 We get y which is positive so this tells e the graph is above 7 the -ais. So ow we kow the graph looks like the followig: Now we eed to take care of the iddle part of the graph. I betwee the two vertical asyptotes a ratioal graph will always look like oe of the followig:

6 Sectio.6 Notes Page 6 If we look at where our itercepts are that will help us choose oe of the followig. The oly graph above that would fit the itercepts we are already plotted would be the third graph, so the followig will be the our copleted graph: EXAMPLE: Fid the itercepts, asyptotes, ad graph of 4 y. 5 ( ) Let s factor this first: y. We should always factor soethig like this to see if aythig ( 3)( 5) cacels. Nothig does o this oe so we will proceed to fid the iforatio. First we will fid the -itercept by settig the top equal to zero: ( ) 0. We will get = 0 ad = -. We will write our aswer as (0, 0) ad (-, 0). Sice (0, 0) is a -itercept the autoatically this is our y-itercept as well. To fid the vertical asyptote, set the botto equal to zero: ( 3)( 5) 0 We get = 3 ad = -5. For the horizotal asyptote, otice that the highest power o top is the sae as the highest power o the a botto, so we kow that y. Here the a ad b, so the horizotal asyptote is y. b Now we are ready for the graph. First plot the itercepts. The use dotted lies to draw i the asyptotes. There will be a horizotal lie through y =. The other asyptotes are = 3 ad = -5. These will be vertical lies goig through 3 ad 5 o the -ais. Now we are ready to draw the graph. Let s look at the part of the graph to the left of = -5. We eed to decide whether the graph will be above or below the horizotal asyptote. We actually do ot eed to use test poits for this. Notice that our oly itercepts are betwee the two vertical asyptotes. This eas this is the oly play the graph crosses the -ais. So whe we look at the part of the graph where is less tha -5 ad sice there are o itercepts there I kow the graph has to be above the horizotal asyptote. The sae reasoig ca be said about the part of the graph where is greater tha 3. There are o -itercepts i the part of the graph where is greater tha 3, so I kow this part of the graph will also be above the horizotal asyptote. Now let s look betwee the two vertical asyptotes. Reeber there are oly four types of graphs that could appear here:

7 Sectio.6 Notes Page 7 By the way the itercepts are located this tells e that oly the first two graphs would work. How ca we tell which oe it is? We eed to pick a test poit. It ca be ay uber betwee -5 ad 3. I will test = -3. I will ( 3) 4( 3) 7 put this i for i the origial equatio: y. Sice this uber is egative that ( 3) ( 3) 5 30 eas whe is -3 the graph should be below the -ais. This is oly goig to happe with the secod graph above (upside dow parabola). So ow we ca fiish our graph: EXAMPLE: Fid the itercepts, asyptotes, ad graph of y. 4 Let s factor this first: y. Now we will fid the -itercept by settig the top equal to zero: ( )( ) 0. We are doe. We will write our aswer as (0, 0). Agai sice (0, 0) is a -itercept the autoatically this is our y-itercept as well. To fid the vertical asyptote, set the botto equal to zero: ( )( ) 0 We get. For the horizotal asyptote, otice that the highest power o top is less tha the highest power o the botto, so we kow that the horizotal asyptote is autoatically y = 0.

Sectio.6 Notes Page 8 Now we are ready for the graph. We oly have oe poit to plot this tie. The we have our dotted lies to draw i the asyptotes. There will be a horizotal lie through y = 0.

8 Sectio.6 Notes Page 8 Now we are ready for the graph. We oly have oe poit to plot this tie. The we have our dotted lies to draw i the asyptotes. There will be a horizotal lie through y = 0. The other asyptotes are. These will be vertical lies goig through - ad o the -ais. Now we eed draw the graph. Right ow ot uch iforatio is give. We eed to use test a value that is less tha - ad greater tha - to see if the graph is above or below the -ais. First I will test = -3. We will put this i 3 3 for i the origial equatio: y. We get a ( 3) 4 5 egative uber so I kow the graph is below the -ais at this poit. I also eed to test a poit greater tha. I will 3 3 choose = 3. You will get y. So I kow that the graph ust be above the -ais. What about the part of the graph betwee the two vertical asyptotes? We eed to do test poits here as well to deterie which of the four odels the graph resebles. I will test = ad = -. We will ot test those values: y This tells us that whe = the graph is below the -ais. 4 3 Now if we test = - you will get y. So we kow the graph is above the -ais at = -. ( ) 4 3 This tells us that the oly graph that will work i the iddle is the third oe. EXAMPLE: Fid the itercepts, asyptotes, ad graph of 4 y. ( )( ) Let s factor this first: y. Now we will fid the -itercept by settig the top equal to zero: ( )( ) 0. We get. well ca write this as,0. For the y-itercept you would put i a zero for. If you do the you will be dividig by zero, so there is o y- itercept i this case. To fid the vertical asyptote, set the botto equal to zero: = 0. We oly have oe vertical asyptote here.

9 Sectio.6 Notes Page 9 For the horizotal asyptote, otice that the highest power o top is ore tha the highest power o the botto, so we kow that this is o horizotal asyptote. However we eed to fid the oblique asyptote by doig log divisio: We eed to put i a 0 ter sice it is issig. After we do the divisio we ed up with a + 0. This is the equatio of the lie that the graph will follow. We eed to set up 0 our graph by plottig the itercepts ad the asyptotes. We will have oe vertical 0 asyptote at = 0 ad the oblique asyptote will be the lie y =. 4 We see where the graph will cross the -ais. This tells us that the graph will be i these sectio. To the left of the vertical asyptote the graph will follow the lie y = ad tur tur towards (-, 0). The it will follow the vertical asyptote. To the right of the vertical asyptote the graph will follow the vertical asyptote, tur through (, 0) ad the follow the lie y =. EXAMPLE: Fid the itercepts, asyptotes, ad graph of y. 6 (3 )( 4) Let s factor this first: y. Notice that we ca cacel out the 4 fro the top ad botto. ( 4)( 4) Wheever this happes you will have what is called a hole i the graph. Whatever part you ca cacel you wat to set this equal to zero. We will have 4 = 0. Solvig it we get = 4. This is ot a vertical asyptote. There will be a hole i the graph at the value of 4. 3 So the equatio we will ow look at to get the iforatio will be y. 4

10 Sectio.6 Notes Page 0 To fid the -itercept by settig the top equal to zero: 3 0. We get, so we have. 3 3, 0 3(0) For the y-itercept you would put i a zero for. If you do the you will have y. So 0,. (0) 4 To fid the vertical asyptote, set the botto equal to zero: + 4 = 0 ad we get = -4. We oly have oe vertical asyptote here sice the other ter caceled out. We already deteried that there will be a hole at the value of 4. To fid the horizotal asyptote we see the highest power o top is the sae as the highest power o the a 3 botto. So the horizotal asyptote is y. Here the a 3 ad b, so y 3. b So we have draw i the asyptotes ad the itercepts. We otice that there are o -itercepts to the right of = -4. This tells us that the graph ust be above the horizotal asyptote (y = 3). If the graph was below this lie the it would have crossed the -ais but we kow this does ot happe. To the right of the = -4 we otice the graph does cross the -ais so we kow the graph is i this lower portio ad ot above y = 3. Of course if you are usure where the graph should be you ca always use tests poits. You test a poit less tha = -4 ad greater tha = -4. Whe we draw the graph we eed to ake sure that we put a hole whe = 4.

AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY

AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY Dr. Farha I. D. Al Ai * ad Dr. Muhaed Alfarras ** * College of Egieerig ** College of Coputer Egieerig ad scieces Gulf Uiversity * Dr.farha@gulfuiversity.et;