Week 3. Topic 5 Asymptotes

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1 Week 3 Topic 5 Asmptotes Week 3 Topic 5 Asmptotes Introduction One of the strangest features of a graph is an asmptote. The come in three flavors: vertical, horizontal, and slant (also called oblique). Asmptotes are lines that the function gets reall close to. There are more technical definitions, of course. The friendl functions ou ve seen before in math classes, like linear and quadratic, do not have an asmptotes. But other tpes do, so we ll keep coming back to them as we proceed through eponentials, logarithms, polnomials, and rationals. Homework There will be a question or two in the Week homework quiz about asmptotes.

2 Week 3 Topic 5 Asmptotes Reading At the basic level, an asmptote is a line that a graph become nearl identical to, at some point. It s a rather difficult concept to eplain in just a few words, so let s look at each tpe of asmptote separatel. Horizontal Asmptotes A horizontal asmptote is a tpe of long-run behavior, or what the graph does for ver large values of or ver large negative values of. For eample, check out this graph: f()=/(+^)+ = There s a hill in the middle, but when gets reall big, the -values get reall close to. In fact, the further ou go to the right or to the left, the closer the -values get to. Let s zoom in on a portion of the graph to the far right. Notice the scale on both aes. For -values bigger than 0, the -values are all within about 0.0 units of..05 f()=/(+^)+ =

3 Week 3 Topic 5 Asmptotes 3 Let s zoom in even more and go even more to the right..005 f()=/(+^)+ = This graph shows the function and the line from = 0 to = 30. The distance is even smaller. Now the -values are all within of. You could keep zooming in and moving further to the right if ou wanted the function s -values too all be less than , if ou reall wanted to. Since this graph s -values approach the horizontal line =, we sa it has a horizontal asmptote at =. Long-run behavior means we re ignoring what happens near the -ais. For eample, a graph could have more than one horizontal asmptote, one for positive and one for negative Notice that for ver large positive, the -values get close to. For ver large negative, the -values get ver close to.5. Other values of don t matter, onl those far out.

4 Week 3 Topic 5 Asmptotes Notice from the graph above that this function does cross the horizontal asmptote. That s perfectl fine. What gives a graph a horizontal asmptote is that eventuall the -values are reall close to the line. In the graph below, we have a function that crosses the horizontal asmptote infinitel often. But notice when gets bigger, the -values still get closer and closer to the line = f()=sin(5)*e^ = Don t think about an asmptote as a line that the graph doesn t cross. That s not quite right. Think about it as a line that the graph becomes nearl identical to, eventuall.

5 Week 3 Topic 5 Asmptotes 5 Vertical Asmptotes Vertical asmptotes are the same idea, but in the other direction. The complication is that we usuall deal with functions that are in the form f ( ) = or =, so it s probabl easier to think about horizontal and vertical asmptotes as different things. For eample, here are two graphs with vertical asmptotes at =. f()=/(-) = = =e^(-.(+5))*sin() Onl the graph on the left is the graph of a function. The graph on the right doesn t pass the Vertical Line Test. A function graph can t cross a vertical asmptote. The function graphed above on the left is f ( ) =. Notice the domain of this function is all real numbers ecept =. And the vertical asmptote is also at =. We can t put into this function for, but an number close to works just fine. Let s look at a table of values f() 0. = = = 000 Using a number close to makes the denominator ver close to 0. Dividing b a number close to zero makes the whole fraction ver large. So the closer ou get to, the bigger the - values get. Here, the get big in the negative direction, and ou can see that in the graph. If we use values such as.0,.00,.000, we get ver big positive -values.

6 Week 3 Topic 5 Asmptotes Vertical asmptotes in function graphs come about when denominators are equal to zero. Let s sa ou have a function where the denominator is zero when = 0. On the graph, when our -values are close to 0, our -values must either increase (to infinit!) or decrease (to negative infinit!). If ou won t have a zero denominator, ou can t have vertical asmptotes, even if the graph looks like there might be one. For eample, here s the graph of asmptotic, doesn t it? f()=^ f ( ) =. Looks like it might be getting a little But no, this graph has no vertical asmptotes. The arrows indicate it goes up forever, but for an -value, there is a defined non-infinite -value. If we zoom out, it looks like this: f()=^ There are no asmptotes here. A steep graph is not the same as a graph with asmptotes. There must be a line where the -values shoot up to infinit (or down to negative infinit) and ou never cross the line itself. One final note: The word is asmptote. Don t sa asmtope. Ever. If ou hear someone else sa it that wa, shake our head furiousl and shout No!

7 Week 3 Topic 5 Asmptotes 7 Slant Asmptotes (also called Oblique Asmptotes) Asmptotes are alwas lines, but the don t have to be vertical or horizontal. The can be angled too. This graph has an oblique asmptote at +. It also has a vertical asmptote at =. f()=()*( + )/( - ) =+ = These are also eamples of long-run behavior, so it doesn t matter what happens in the middle of the graph, onl for ver large -values. So a function s graph can cross an oblique asmptote, like shown here: f()=3()*(- + )*( + )/( - )^ = This function has a slant asmptote at =. The function crosses it at = = This sort of asmptote isn t ver important. In the last week, of the course, we ll talk about these a little more but as one tpe of long-run behavior of rational functions.

8 Week 3 Topic 5 Asmptotes 8 Practice Problems. Sketch graphs of functions with these properties. Use dotted lines to indicate asmptotes. a. The function has a vertical asmptote at = and a horizontal asmptote at = - b. The function has a slant asmptote at = + and a vertical asmptote at = 0. Identif the asmptotes (horizontal, vertical, and slant) in each of these graphs. Give each asmptote as an equation. a b

9 Week 3 Topic 5 Asmptotes 9 Practice Problem Solutions. There are man possible graphs. Just be sure ours fits all the criteria. You don t have to come up with equations, just draw the graphs. a. The function has a vertical asmptote at = and a horizontal asmptote at = - f()=/(-)- = - = - - b. The function has a slant asmptote at = + and a vertical asmptote at = 0 f()=(^++)/ = + = Identif the asmptotes (horizontal, vertical, and slant) in each of these graphs. Give each asmptote as an equation = f()=-/( - )^+ = -7-9 = - f()=( + )*( - )/( + ) f()= Vertical: = Horizontal: = Vertical: = - Slant: =