section10: graph of 1/f(x)this is a y-axis type of transformation so the x-coordinates stay the same while the y coordinates are changed.
for example a
point (u,v) will be turned into (u, 1/v) if v
isn't equals to zero.
if it is
zero then that point will be undefined and
x=u will become an asymptote.So by this all the x-intercepts will become vertical asymptotes.
The
vertical asymptotes in the y=f(x) graph will become the
new x-intercepts,
while the
horizontal asymptotes will undergo a transformation that will make it
y=(1/k) if the asymptote is y=k.
in the graph above the x-intercepts will become undefined and therefore asymptotes as seen below.
the
minimum point of your f(x) graph will also become the
maximum point on the next graph. the new vertical asymptotes will be x=1 and x=5.
you will notice that the derived shape is FAR from the original shape and it may seem really alien to you. but the simplest way to derive the shape would be to
sub in value at the correct points. for example, take the points near x-coordinate 3. for
values lesser and lesser than 3, the y-coordinate will
become smaller. likewise for
values more and more than 3. hence you will get the negative parabola.
For the case of y=x-2, the same thing happens, the x-intercept becomes the asymptote for the graph as seen below. And the rest of the points just follow and become (u , 1/v). The whole graph becomes curved as seen below. Simple right!
so basically this alien-y section can be summarised as follows:
for coordinates:point
(u,v) will be turned into
(u, 1/v) if v
isn't equals to zero.
if it is
zero, it will be
undefined and x=u will become an asymptote.
for asymptotes:
vertical asymptotes will become the
new x-intercepts.
horizontal asymptote y=k will
become y=(1/k).
hoho, time to try out your newly acquired knowledge!
question: transform the following y=g(x) graph to obtain y=2/g(x).
now choose!
