## anonymous 3 years ago how to graph general rational functions? like y=2x/x^2-1

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1. anonymous

first, you'll need to know the asymptotes. in this case, as$$x^2 -1 \neq 0$$ $$x= \pm 1$$ is the asymptotes.

2. anonymous

then you'll need to know the max/min points. use differentiation to find them and plot the points.

3. anonymous

so like two and 1? also how do you know or like get if it has a 0 or not?

4. anonymous

$$y=2x(x^2-1)^{-1}$$ $$y'=2(x^2-1)^{-1} -2x(x^2-1)^{-2}$$

5. anonymous

how did you get the two and 1?

6. anonymous

umm cuz there's a 2 and a 1? i was just trying but i knew it probably wasn't that easy

7. anonymous

nope. you'll have to use the global max/min test to find the points. at the max points, their gradients=0, so y'=0.

8. anonymous

is it because it's 2x it's zero? like cuz of the x or if not then plz explain that that global max/min test is cuz i don't get it

9. anonymous

at the max point, |dw:1355369628774:dw|

10. anonymous

wait. is this a precalculus question?

11. anonymous

? that looks higher than 0 aka the 0 on the y axis unless that has somethin to do with the gradient cuz idk what that is and yes this is a precalc question though my algebra book has it in there.

12. anonymous

I put the question here and like in a few other math places cuz i thought maybe someone would answer my question cuz when i put it in only one i was waiting for like ten minutes and nobody answered

13. anonymous

okay, then you might want to know that it should look like this.|dw:1355369865285:dw| without calculus, you'll just have to memorise the shape and plot points.

14. anonymous

well i'm in precalc but i don't get what my teacher does so that's why i'm here. so just try to explain like if i'm five or really dumb and give reasons for what your doing like if i said why is 2+2=4? then say because if you had 2 pencils and you got two more you get 4! and you get 2+2 because adding is like getting more to what you have. see where i'm goin with this? so yes your graph is uh helpful but how and why did you get there?

15. anonymous

okaay, we both know that a fraction cannot have a zero at the bottom, right, or it won't exist. so, your fraction, y also cannot have a value of 0 for $$x^2-1$$ or it doesn't exist.

16. anonymous

okay so far so good next?

17. anonymous

how ever, $$x^2-1$$ can have a value very near 0, but not zero. the asymptotes are the lines where the fraction starts to get very very near infinity(it doesn't exist)

18. anonymous

i.e. the graph at those lines will go into infinity. (as shown in the pic on top)

19. anonymous

umm real quick what is x was 1 then it would be 1-1 which is 0? or is it because it's 1 that that's where you graphed it?

20. anonymous

since $$x^2- 1$$ cannot be zero, but can be very near zero, we need to find that limit. so, we let $$x^2- 1=0$$ and solve for x. then those values of x are the values that x can be very near to, but equal to.

21. anonymous

OH! okay! i understand!

22. anonymous

glad you do. now, we need to know if there is a point of importance. Drawing the graph for y=2x and y=x^2 -1, we can see that there is a point that both of them have, (0,0)

23. anonymous

so, the graph will near two asymptotes, one key point, and you just need two points, beside the key point. i.e. when x=-0.1 and x=0.1 so that we know how the graph turns

24. anonymous

okay i am really super sorry to bring this up but um i thought there couldn't be any zeros? or did you just round it to zero because the number it actally is 0.1 is really close to zero?

25. anonymous

as long as the bottom is not zero, then there could be a zero. and yes, i chose 0.1 as it is very close to the key point, x=0.:)

26. anonymous

OH so if x=0 then 0^2-1 would =-1 and that's where a vertical asyptote in your graph is!!!

27. anonymous

sorry, i guess i didn't explain what the key point is. the key point (0,0) is the point of inflexion, where the gradient(the shape) of the graph will change. -1 is obtained after $$x^2-1=0$$

28. anonymous

okay so now what?

29. anonymous

so, what did you get when x=-0.1 and 0.1?

30. anonymous

x=.01and .2