IM 9 Advanced Rational Functions: Holes and Asymptotes Investigation We have already seen that nearly all rational functions have a vertical asymptote. This is because they are “undefined” when the denominator becomes 0. On the TABLE of your calculator this is displayed with an ERROR message. However, rational functions can be undefined even when the graph does not show an asymptote! Follow the investigation below to see how. Holy Cow! A hole in the graph?!?!?: Recall that a vertical asymptote at 𝑥 = 𝑘 is a discontinuity !
(break in the graph) caused by a factor of !!! appearing in the function. If the factor instead looks !!!
like !!! then the discontinuity becomes a hole at 𝑥 = 𝑘. Ummmmm, you totally lost me. What?!?!? ! ! !!!!!
Example 1: Consider the function 𝑓 𝑥 = (!!!)(!!!). 1) Where will the function be undefined? Why? 2) Using your GDC, graph f(x) on the standard viewing window. How many vertical asymptotes are present in the graph? Does this answer conflict with your answer to part 1? 3) Now look at the TABLE on your GDC. Where does the ERROR message appear? What is this consistent with? What is this inconsistent with? To resolve this conflict we’ll look at the factored form of the rational function, f(x). 𝑥 ! − 5𝑥 + 6
(𝑥 − 2)(𝑥 + 3)
𝑓 𝑥 =
=
(𝑥 − 2)(𝑥 − 1) (𝑥 − 2)(𝑥 − 1)
The factor 𝑥 − 2 appears in both the numerator and the denominator. If we cancel that factor we’re left with: (!!!)
𝑓 𝑥 = (!!!). Graph this function on the same graph you used above. What do you notice? You probably noticed is that the two graphs are identical. This is because part of the factored (!!!)(!!!)
!!!
(!!!)
form of 𝑓 𝑥 = (!!!)(!!!) is !!! , which is equal to 1. As such, if we start with 𝑓 𝑥 = (!!!) and !!!
multiply by !!! we are only multiplying by 1, so the values on the graph won’t change! The two graphs will look the same. However, we are introducing a new place where the graph is undefined. This is how we get a “hole” in the graph – a place where the function is undefined (ERROR) but where there is no vertical asymptote. Unfortunately, a graphing program will not show holes, so here is a graph of 𝑓(𝑥) showing all of its features. Notice there is an open circle on the point 2, −1 . To sum things up, there is a vertical asymptote at 𝑥 = 1, and a hole at 𝑥 = 2. But how do we know the exact location of the hole? This can be found by plugging in 𝑥 = 2 to the “look-­‐alike” function, or the simplified function: 𝑓 𝑥 = (!!!)
(!!!)
Notice also that the graph of 𝑓 𝑥 = (!!!)(!!!)
has a horizontal asymptote at 𝑦 = 1. Recall, making (!!!)(!!!)
the values of 𝑥 very large positively and negatively best sees this. The end behavior of the graph is as 𝑥 → ∞ 𝑦 → 1 and as 𝑥 → −∞ 𝑦 → 1. This gives us the horizontal asymptote at 𝑦 = 1. Example 2: Determine any asymptotes and holes of 𝑓 𝑥 =
!!!
! ! !!!!!
. Solution: 1) Factor the numerator and denominator completely: 𝑓 𝑥 =
!!!
(!!!)(!!!)
. 2) Find x values that will make the function undefined. In this case it will be x = -­‐1 and x = -­‐2 3) Determine, by canceling any common factors in the numerator and denominator, if these !
x-­‐values will be asymptotes or holes. The simplified, “look-­‐alike”, form of f(x) is 𝑓 𝑥 = !!!, so we will have a vertical asymptote at 𝑥 = −2 and a hole when 𝑥 = −1. !
4) Find the location of the hole by plugging 𝑥 = −1 into the “look-­‐alike” form of f(x), 𝑓 𝑥 = !!! . In the “look-­‐alike” function we get 𝑓 −1 = 1, so the coordinate of the hole is −1, 1 . 5) Finally, to see the horizontal asymptote, investigated the end behavior of the function. As 𝑥 → ∞ 𝑦 → 0 and 𝑥 → −∞ 𝑦 → 0. So there is one horizontal asymptote at 𝑦 = 0. ! ! !!!!!
Example 3: Determine any asymptotes and holes of: 𝑔 𝑥 = ! ! !!!!!. (!!!)(!!!)
Solution: By factoring the numerator and denominator completely, 𝑔 𝑥 = (!!!)(!!!), you should find a vertical asymptote at 𝑥 = −4 and a hole when 𝑥 = −2. The coordinate of the hole is (−2,0). Also, there is a horizontal asymptote at 𝑦 = 1. Be sure this agrees with YOUR OWN findings. Exercises (finish for homework): (!!!)(!!!)(!!!)
1. Let 𝐹 𝑥 = (!!!)(!!!) a. What x-­‐values are not part of the domain of 𝐹 𝑥 and why? b. Does the graph of 𝐹 (𝑥) have any vertical asymptotes? If so, write an equation describing each one. c. Does the graph of 𝐹 (𝑥) have any holes? If so, find the coordinates of the hole(s). d. Does 𝐹 (𝑥) have any horizontal asymptotes? How do you know? e. What are the zeros of 𝐹 𝑥 ? f. With the help of your calculator, sketch a graph of 𝐹 𝑥 . (!!!)(! ! !!!!)
2. Let 𝐺 𝑥 = (!!!)(! ! !!!!). Answer the same questions as above for 𝐺 𝑥 . a. What x-­‐values are not part of the domain of 𝐺 𝑥 and why? b. Does the graph of 𝐺 (𝑥) have any vertical asymptotes? If so, write an equation describing each one. c. Does the graph of 𝐺 (𝑥) have any holes? If so, find the coordinates of the hole(s). d. Does 𝐺 (𝑥) have any horizontal asymptotes? How do you know? e. What are the zeros of 𝐺 𝑥 ? f. With the help of your calculator, sketch a graph of 𝐺 𝑥 . 3. Write your own formula for a rational function that has two vertical asymptotes but no holes. Check your answer with a graph. 4. Write your own formula for a rational function that has two holes but not vertical asymptotes. Check your answer with a graph. 5. Write your own formula for a rational function that has a vertical asymptote at 𝑥 = −2 and a hole at 3,4 . Check your answer with a graph. 6. Write your own formula for a rational function that looks like a line everywhere except at 𝑥 = 3. Check your answer with a graph. 7. Find all holes, asymptotes (vertical and horizontal) and ! ! !!! ! !!!
intercepts of 𝑦 = ! ! !!!!! . Using the information you’ve gathered and your calculator as a guide, sketch a graph of 𝑦=
! ! !!! ! !!!
! ! !!!!!
.