## sasogeek Group Title how to find the coordinates of holes in a rational function.... one year ago one year ago

1. amistre64 Group Title

what happens when you remove a part of a line or curve?

2. bahrom7893 Group Title

see where the function is discontinuous

3. sasogeek Group Title

how do i find that if i haven't graphed it?

4. amistre64 Group Title

5. amistre64 Group Title

what happens when you remove a part of a line or curve?

6. amistre64 Group Title

|dw:1348750115514:dw|

7. sasogeek Group Title

okay, that part is either missing or doesn't satisfy the function....

8. amistre64 Group Title

now relate this to removing something from a fraction; what enables us to cancel things top and bottom?

9. sasogeek Group Title

if they're like terms of can be factored?

10. amistre64 Group Title

correct, so any factor that creates a zero in the denominator of a rational function; if it can be removed from the equation, it creates a hole

11. sasogeek Group Title

ok so for example, $$\huge \frac{3x^3}{x^2-1}$$ my hole will be -1,+1 ? same as the asymptotes?

12. amistre64 Group Title

a hole is not "the same" as an asymptote. they have similar effects on the equation (they zero out a denominator); but a hole can be removed. an asymptote cannot.

13. amistre64 Group Title

$\frac{3xxx}{(x+1)(x-1)}$ there are no common factors in this; so no factors can be removed; therefore no holes are created

14. sasogeek Group Title

I'll need more practice on this but thanks :) I'll just state that there's no holes xD

15. amistre64 Group Title

more practice is good ;)

16. sasogeek Group Title

to be clear though, can you give me an example function that has holes?

17. amistre64 Group Title

$\frac{(x+2)(x-3)}{x(x-3)(x+7)}$

18. amistre64 Group Title

can you tell me all the bad xs?

19. sasogeek Group Title

x=3 ?

20. amistre64 Group Title

x=3 is one of the bad xs, yes and since it can be removed from the setup; it creates a hole at x=3

21. sasogeek Group Title

so basically since you can remove that from the function, that makes a hole.... now it's even clearer :) i think i'm getting it :) not quite there yet though :P

22. amistre64 Group Title

$y=\frac{(3+2)\cancel{(x-3)}}{3\cancel{(x-3)}(3+7)}$ $y=\frac5{30}=\frac16$ so the coordinate of the hole is (3, 1/6)

23. sasogeek Group Title

you said it is one of the holes... are there more?

24. amistre64 Group Title

can we remove any of the other factors from the equation? if not, then there are no more holes to be found. the rest of the zeros in the denominator would create vertical asymptotes

25. sasogeek Group Title

okay :) thanks. can i assume then that if the zero in the denominator cancels out something in the numerator, it's a hole, if not, it is a vertical asymptote?