We'll actually use the graph of
the basic rational function, f of x equals one divided by x,
which is graphed here in gray,
to graph g of x equals one divided by
the quantity x plus one minus two.
For a quick review, let's look at an animation.
We have the graph of f of x equals one divided by x,
and now let's see how the value of
c and d affect the graph of g of x.
Notice how when c is positive,
where we have the quantity x plus a constant
in the denominator, the graph is shifted left.
And when c is negative, we have x minus a quantity
in the denominator, the graph is shifted right.
So if c is negative, the graph is shifted right c units.
And if c is positive, the graph is shifted left c units.
Now, let's see how d affects the graph.
When d is positive, the graph is shifted up d units.
And when d is negative, the graph is shifted down d units.
We can summarize these two types of translations
by saying that if c is negative,
the graph is shifted right c units.
And we can write g of x in the form
of f of the quantity x minus c.
If c is positive, the graph is shifted left c units,
and we can write g of x in the form
of f of the quantity x plus c.
And if d is less than zero or negative,
the graph is shifted down d units.
Then we can write g of x in the form of f of x minus d.
And if d is positive, the graph is shifted up d units,
and we can write g of x in the form of f of x plus d.
So going back to our example, notice how because we have
x plus one in the denominator and a minus two on the end,
we can say that g of x is equal to f of,
instead of x in the denominator, we have x plus one,
so it'd be f of the quantity x plus one, and because of
the minus two on the end, we'd have minus two.
So this tells us that c is equal to positive one
and d is equal to negative two.
So because c is equal to one and d is equal to negative two,
we can say that g of x is f of x
shifted,
again because c is one or because we have
x plus one in the denominator,
the graph is shifted left one unit.
And because d equals negative two
or because we have minus two on the end,
it's also shifted down two units.
So now what we can do to graph g of x is shift
these six key points of the basic rational function
f of x equals one divided by x
left one unit and down two units.
So starting with this point here,
we'll go left one, down two,
this point, left one, down two,
and this point, left one, down two.
And now for these points, left one, down two,
left one, down two, and left one, down two.
And let's also shift the horizontal and vertical asymptotes.
Starting with the vertical asymptote,
we can shift it left one unit, which would bring it to here.
Notice how if we shift it down two units,
it's not going to change the line.
Now for the horizontal asymptote,
we'll shift it down two units.
Notice by shifting it left one unit,
it does not change the horizontal line.
And now we have enough information
to make a nice graph of g of x.
It passes through these three points
and approaches the asymptotes...
and passes through these three points
and approaches the asymptotes.
I hope you found this helpful.