Welcome to a lesson
on how to graph a translation of the basic
or apparent rational function,
f of x equals one divided by x.
So this video does assume we're familiar
with the basic rational function,
f of x equals one divided by x, which is graphed here.
And we're first asked to graph g of x,
which equals one divided by the quantity x plus three,
as well as g of x equals one divided by x minus two.
Let's begin by looking at an animation
showing translations of f of x equals one divided by x.
So here we have the graph of the parent function
f of x equals one divided by x,
and we're going to look at how the value of c and d
affect the graph of the parent function.
Notice how c is a value added to x in the denominator
and d is the constant added on the end.
Let's see what happens when we change the value of c.
Notice when c is positive,
the graph is shifted to the left,
which means we'd have x plus a constant
in the denominator, so if c is positive the graph
is shifted left.
Notice when c equals four, the graph is shifted
exactly four units to the left,
and it should make sense because if we wanted the y values
of the parent function and the translated function
to be the same, because we have quantity x plus four
in the denominator of the translated function,
we'd have to use input direct values
that are four or less to get the same output
to our y values.
And notice how when c is negative,
or we have x minus a constant in the denominator,
the graph is actually shifted to the right.
So notice here when c is equal to negative three,
we'd have the quantity x minus three in the denominator,
and the graph is shifted exactly one, two, three units
to the right.
And again, this should make sense because
to want these two functions to have the same y values
or function values, the inputs or x values
of the translated function would have to be three
units larger to get the same outputs or y values,
because the function would be one divided
by the quantity x minus three.
So again, if c is negative, the graph is shifted right.
If c is positive, the graph is shifted left.
Now let's set c back to zero,
and determine how d affects the graph
of the parent function.
Notice how if d is positive,
the graph is shifted up d units,
and if d is negative, the graph is shifted down d units.
So if we have minus a constant on the end,
the graph is shifted down,
if we have plus a constant, the graph is shifted up d units.
So going back to our lesson to summarize
we can say if our function g of x is equal
to f of the quantity x minus c,
then the graph is shifted right c units.
If g of x is equal to f of the quantity x plus c,
the graph is shifted left c units,
and if g of x equals f of x minus d,
the graph is shifted down d units,
and if g of x equals f of x plus d,
the graph is shifted up d units.
So notice how here we can say that g of x
is equal to, if we replace x with x plus three,
the functions would be the same,
so we could say g of x equals f
of the quantity x plus three.
So because we're adding three to the input x,
this is going to shift the graph
of f of x left three units.
So we'll say g of x is f of x
shifted left three units.
So now to graph g of x, we'll use these key points
on the parent function and shift them left three units,
and we'll also shift the vertical asymptote
left three units.
Let's first shift the points.
So let's shift the point one comma one
left three units which would be here.
This point left three units which would be here,
and this point left three units which would be here.
And now I'll shift these three points left three units,
so one, two, three,
one, two, three,
and one, two, three.
For the parent function the vertical asymptote
is the y axis which is x equals zero.
If we shift that left three units,
the vertical asymptote will be x equals negative three,
which is here.
If we shift the horizontal asymptote three units left,
it will be the same horizontal line,
which is the x axis.
This is all we need in order to make a nice graph
of g of x.
It'll pass through these three points,
and approach the vertical and horizontal asymptotes,
and will also pass through these three points,
and approach the horizontal and vertical asymptotes.
So this is a graph of g of x,
which equals one divided by the quantity x plus three.
Notice how knowing the translation of the parent function
saves a lot of time rather than making a table of values.
Now let's look at g of x equals one divided by x minus two.
Because f of x equals one divided by x,
we can say that g of x is equal to f of x minus two,
which means g of x is f of x shifted down two units.
And this should make sense because if we take
the function values or y values of f of x,
and subtract two, it will shift each point down two units.
So let's take these six points and shift them
down two units.
So the point 1/2 comma two down two units would be here,
this point down two units would be here,
this point down two units would be here.
Now for these three points, down two units,
down two units, and down two units.
Now here because we have a vertical shift,
we will now shift the horizontal asymptote down two units.
So right now the horizontal asymptote is the x axis
which is y equals zero.
The new horizontal asymptote will be y equals negative two.
Should be this horizontal line here.
And of course if we shift the vertical axis
down two units it won't change,
it'll still be the y axis.
So now we can easily graph g of x equals one divided
by x minus two.
It'll pass through these three points,
approaching the asymptotes,
as well as these three points, and approach the asymptotes.
Let's take a look at two more examples.
Here we have g of x equals one divided by x plus one.
Which means we can say that g of x
is equal to f of x plus one,
so because of the plus one we know that g of x
is f of x shifted up one unit.
So we'll shift each point of the parent function
up one unit, as well as shift the horizontal asymptote
up one unit.
Let's first shift the points, so up one unit
here, here, and here.
As well as here, here, and here.
And we'll shift the horizontal asymptote
from the x axis where y equals zero to y equals one.
Shifting the vertical asymptote up one unit
does not change its location, it is still the y axis.
So the graph of g of x equals one divided by x plus one
would have one piece here,
and one piece here.
And for our last example we want to graph
g of x equals one divided by the quantity x minus two.
Notice here we can say that g of x is equal to
f of the quantity x minus two,
and because we have the quantity x minus two here,
g of x is the graph of f of x shifted right two units.
So we'll take the six key points on the parent function
and shift each of them right two units.
So this point right two units,
this point right two units,
and this point right two units.
This point right two units,
this point right two units,
and this point right two units.
And we'll also shift the vertical asymptote
right two units.
So instead of x equals zero,
we'd have x equals two, which is here.
And if we shift the horizontal asymptote
right two units it doesn't change.
It's still the x axis.
And now we can easily graph g of x equals one
divided by the quantity x minus two.
It'll pass through these three points,
and approach the asymptotes,
and pass through these three points
and approach the asymptotes.
So by knowing the translations of parent functions
we should be able to more easily graph
more complicated functions.
I hope you found this helpful.