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sasogeek
how to find the coordinates of holes in a rational function....
what happens when you remove a part of a line or curve?
see where the function is discontinuous
how do i find that if i haven't graphed it?
lets start with big concepts and then refine them for this
what happens when you remove a part of a line or curve?
|dw:1348750115514:dw|
okay, that part is either missing or doesn't satisfy the function....
now relate this to removing something from a fraction; what enables us to cancel things top and bottom?
if they're like terms of can be factored?
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
ok so for example, \(\huge \frac{3x^3}{x^2-1}\) my hole will be -1,+1 ? same as the asymptotes?
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.
\[\frac{3xxx}{(x+1)(x-1)}\] there are no common factors in this; so no factors can be removed; therefore no holes are created
I'll need more practice on this but thanks :) I'll just state that there's no holes xD
more practice is good ;)
to be clear though, can you give me an example function that has holes?
\[\frac{(x+2)(x-3)}{x(x-3)(x+7)}\]
can you tell me all the bad xs?
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
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
\[y=\frac{(3+2)\cancel{(x-3)}}{3\cancel{(x-3)}(3+7)}\] \[y=\frac5{30}=\frac16\] so the coordinate of the hole is (3, 1/6)
you said it is one of the holes... are there more?
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
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?