We're asked to use the graph of
the basic rational function f of x equals one divided by x
to graph the function g of x equals
one divided by the quantity x minus two plus three.
Looking at the form of g of x,
we should recognize that g of x
is a translation of the basic rational function
f of x equals one divided by x.
Let's begin by looking at an animation.
So in blue we have the graph of
f of x equals one divided by x,
and now we're going to look at the graph of g of x
and see how the value of c
and the value of d affect the graph.
Notice right now both c and d are zero,
and therefore f of x is equal to g of x.
So let's see what happens to the graph of g of x
if c increases.
Notice when c is positive,
the blue graph is shifted left,
and it's shifted left c units.
So notice here, when c is three, the function would be
g of x equals one divided by the quantity x plus three
which shifts f of x left three units.
So if c is positive, the graph is shifted left,
or when we have x plus a quantity in the denominator.
Now let's see what happens when c is negative.
Notice when c is negative,
we would have x minus a quantity in the denominator,
and the graph is shifted right c units.
So if c is negative, or if we have x minus a quantity,
the graph is shifted right,
and when c is positive, or we have x plus a quantity
in the denominator, the graph is shifted left.
Now let's see how d affects the graph.
So now c is zero, and now we'll change the value of d.
If d is positive, which is the constant on the end,
the graph is shifted up d units.
And if d is negative, the graph is shifted down d units.
To summarize these two translations,
if we write g of x in terms of f of x,
we can say that when c is negative or less than zero,
the graph is shifted right c units,
and g of x would be in the form of
f of the quantity x minus c.
If c is greater than zero or positive,
the graph is shifted left c units
and g of x can be written in the form of
f of the quantity x plus c.
And if d, the constant on the end,
is negative or less than zero,
the graph is shifted down
and g of x would be in the form of
f of x minus d.
And if d is positive, the graph is shifted up
and g of x would be in the form of
f of x plus d.
So looking at g of x, notice how we can write
g of x as g of x equals,
because we have x minus two in the denominator,
notice how that would be f of the quantity x minus two,
because we have plus three on the end,
we'd have plus three.
This indicates that c is negative two
and d is equal to positive three.
So this information tells us that g of x is
f of x shifted
because c equals negative two,
or because we have the quantity x minus two
in the denominator, the graph would be shifted
right two units.
And because of the plus three on the end
or because d is positive three,
the graph is also shifted
up three units.
So now looking at the graph of the basic ration function,
we'll take these six key points
and shift each of them right two and up three units.
So starting at this point here,
right two, up three.
Next point.
Right two, up three.
And the third point here, right two, up three.
These are three points on the graph of g of x.
Now let's go ahead and shift these three points
right two and up three.
So right two, up three.
Right two, up three.
And right two, up three.
Now let's also shift the horizontal
and vertical asymptotes of the basic rational function.
Starting with the vertical asymptote,
if we shift it right two units,
instead of being the y axis, or x equals zero,
it'd be the line x equals two,
which would be here.
Notice how if we shift the vertical line up,
it does not change the line.
Now if the horizontal asymptote
will shift it up three units,
right now it's y equals zero or the x axis,
so it will end up being y equal three here.
Again, notice how for the horizontal asymptote,
if we shift it right two units,
it doesn't change the line.
This is all the information we need
to make a nice graph of g of x.
One part passes through these three points
and approaches the asymptotes.
The other part passes through these three points
and approaches the asymptotes.
So by recognizing that g of x is a translation
of the basic rational function f of x,
we can more easily graph g of x,
rather than making a table of values.
I hope you found this helpful.