You're given
)
, and we want to know what the reciprocal graph would look like. So we'll firstly look at the behaviour of
})
at certain points of interest on
.
The first one would be what happens to
when we have
intercepts on
? For these points we have
=0)
, so we when try to take the reciprocal we get
}=\frac{1}{0})
, which is undefined. But if we look at the value of this function at values very close to this value of
, we see that
is really small, that is
\rightarrow 0)
. When we divide by this really small number, we get a really big number, which goes off to approach infinity,
,
}\rightarrow \infty)
. So we get a vertical asymptote!
That is when we have
intercepts on , we get vertical asymptotes on the reciprocal graph.Now lets look at what happens to the rest of the curve. We know that it has to approach our asymptotes we've found, but for this to happen the curve must turn back at some point. If we say take a local minimum at
))
, then on our reciprocal graph we get
})
. The values of the function either side of this were originally larger than our
value at the turning point, but now since we are taking the reciprocal they'll be smaller and decreasing.
So our local minimum will turn into a local maximum with the value being the reciprocal of our original value. The same goes for local maximums becoming local minimums. We should note for any stationary points which have a
value of
will be in the same place on the reciprocal graph, where as if the
is less than
then it will become greater than
and vice-versa.
We should also observe the behaviour of
as it goes off to
and
. If it starts getting really large, then the reciprocal will become really small and vice versa.
Our last piece of information we would need to show would be the
intercept, which is similar to the other points on the graph, with it having the same
value but
over the
value.
Spoiler
So in your question the curve has an
intercept at
, so this will become a vertical asymptote at
. The first turning point is a local maximum at
)
will become a local minimum at
)
, the second turning point is a local minimum at
)
, will become a local maximum at the same point. Also as
and
,
\rightarrow \infty)
, so
}\rightarrow 0)
.
Draw these in lightly and you should be able to see how the curve should connect to itself.
EDIT: Tried to graph it on cas using the equation, turns out the turning points and such aren't exactly at those points on the graph, because there isn't an actual curve that goes through those points, so using the points above as a 'guide' to where the features should be rather than being exactly on those points.
EDIT2: Tried to draw it in paint, so it's not great but it'll do. On the left is what features you'd use to see the shape and on the right is the actual curve.
EDIT3: The actual curve on the left of the asymptote would probably be a lot sharper than that.
EDIT: Edited, added stuff e.t.c.
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