Welcome to a lesson on how to graph
the reflection of the basic rational function,
or parent rational function, f of x equals one divided by x.
Here we're asked to use the graph
of f of x equals one divided by x,
which is graphed here in gray,
to graph g of x equals f of negative x
and g of x equals negative f of x,
or the opposite f of x.
Notice here the relationship is,
we change the sign of the x values or inputs of f of x
in order to determine the function values of g of x.
And here we change the sign of the outputs,
or y values of f of x,
in order to determine the function values of g of x.
Let's see if we can understand why one of these
is a reflection across the vertical, or y-axis,
and another is a reflection
across the horizontal, or x-axis.
The graph of g of x equals f of negative x
is the graph of f of x reflected across the y axis.
Again, looking at this notation here,
we're changing the sign
of the x values, or inputs, of f of x
in order to determine the function values,
or y values, of g of x.
So, for example, if this is one point on f of x,
in order to find the corresponding point on g of x,
we need to change the sign of the x coordinate.
If we change the sign of the x coordinate,
that would reflect the point across the y axis,
or vertical axis, giving us this point here.
So this is the reason why,
when we have g of x equals f of negative x,
the graph of g of x is a reflection
across the vertical, or y-axis.
And then if g of x equals the opposite of f of x,
or negative f of x,
the graph of g of x is the graph of f of x
reflected across the x-axis.
Here we're changing the sign of f of x,
which are y values,
in order to determine the function values of g of x.
So again, if this point is the point on our function f of x,
and we change the sign of f of x,
or change the sign of the y coordinate,
it's going to reflect that point
across the horizontal, or x-axis,
giving us the point x comma the opposite of f of x.
And this is the point that will be on our function g of x.
So going back to our two examples,
for our function g of x here,
we'd be changing the sign of the x coordinates
of the points on the parent function,
which is a reflection across the vertical, or y-axis.
So let's say g of x
is f of x
reflected
across the vertical, or y-axis.
So we'll use these six key points on the parent function,
reflect them across the y-axis, and then graph g of x.
So again, this is going to be a reflection
across the y-axis, or vertical axis,
so these points are going to be on the left.
These points are going to be on the right.
This first point is half a unit to the right.
If we reflect it across the vertical, or y-axis,
it'll be half a unit to the left of the vertical axis.
This point will be one unit to the left.
This point will be two units to the left.
Notice how the y values, or function values,
remain the same.
And now we'll go ahead and reflect these three points.
This point is now two units to the left,
reflecting across the vertical axis.
It'll be two units to the right, which would be here.
This point will be one unit to the right.
This point will be half a unit to the right.
If we reflect the vertical and horizontal asymptotes,
they would remain the same,
because this is on the vertical axis,
and this is on the horizontal axis.
And now we now the graph of g of x equals f of negative x
passes through these three points
and approaches the asymptotes,
and passes through these three points
approaching the asymptotes.
Next we have g of x equals
the opposite of f of x, or negative f of x.
Here we'd be changing the function values, or y values,
of f of x to determine the function values of g of x.
If we change the sign of the y coordinates of f of x,
it would reflect the graph
across the x-, or horizontal, axis.
So let's go ahead and clear this.
Let's state the reflection:
g of x
is f of x
reflected
across
the horizontal, or x-axis.
So again, in order to graph g of x,
we'll change the sign of the y coordinates
of these six points,
which will again reflect the graph
across the horizontal axis.
So this piece of the graph will be below the x-axis.
This piece will be above the x-axis.
So this point is two units above the x-axis.
After the reflection, it will be two units below.
This point is one unit above.
After the reflection, it will be one unit below.
This point is half a unit above.
After the reflection, it will be half a unit below.
This point will be two units above the horizontal axis.
This point will be one unit above.
And this will be half a unit above.
If we reflect the asymptotes across the horizontal axis,
again they won't change.
And now we can graph g of x equals negative f of x.
It'll pass through these three points
approaching the asymptotes,
as well as these three points and approach the asymptotes.
And what you might recognize here
is that the graph of g of x equals negative f of x
and the graph of g of x equals f of negative x
are exactly the same, but for different reasons.
And let's verify this again using some graphing software.
Here's a graph of the basic function, or parent function,
f of x equals one divided by x.
Now let's look at the graph of f of negative x,
which gives us the blue graph,
because we changed the sign of the inputs, or the x values.
This is the reflection across the vertical, or y-axis.
And now it's also a graph of the opposite of f of x,
which gives us the green graph,
which is the same as the previous graph, the blue graph.
But here, because we changed the sign of f of x,
or changed the sign of the y values of f of x,
this is the reflection across the horizontal, or x-axis.
In this case, though,
the horizontal and vertical reflections
result in the same graph.
I hope you found this helpful.