Video Transcript
Sketch the graph of the function 𝑓
of 𝑥 is equal to one divided by 𝑥 plus two minus one and then find the horizontal
asymptote of 𝑓 of 𝑥. Find the vertical asymptote of 𝑓
of 𝑥.
In this question, we’re given a
function 𝑓 of 𝑥 which is equal to one divided by 𝑥 plus two minus one. And we want to determine both the
horizontal asymptote of the graph of this function and the vertical asymptote of the
graph of this function. And we’re asked to do this by
sketching the graph of the function. So to do this, let’s start by
looking at the graph we’re asked to sketch, one divided by 𝑥 plus two minus
one. And one way of doing this is to
note 𝑓 of 𝑥 is very similar to the standard reciprocal function 𝑔 of 𝑥 is equal
to one divided by 𝑥. In fact, we can transform the
reciprocal function onto the function 𝑓 of 𝑥 by using the following
transformations.
We note that 𝑔 of 𝑥 plus two
minus one is equal to 𝑓 of 𝑥. We find these values directly from
the given function 𝑓 of 𝑥. We can then recall if we add two to
the input values of the function, then we’re going to translate the curve
horizontally two units to the left. Similarly, if we subtract one from
the outputs of the function, then we’re going to translate the curve vertically one
unit down. Therefore, we can sketch the curve
𝑦 is equal to 𝑓 of 𝑥 by sketching the curve 𝑦 is equal to one over 𝑥 and then
applying these two transformations.
To do this, let’s start by
recalling what the graph of the reciprocal function 𝑦 is equal to one over 𝑥 looks
like. Its shape looks like the
following. As the values of 𝑥 approach ∞, the
outputs of the function approach zero from the positive direction. And as the values of 𝑥 approach
negative ∞, the outputs of the function approach zero from the negative
direction. The 𝑥-axis is a horizontal
asymptote as shown. Similarly, as the values of 𝑥
approach zero from the positive direction, the outputs of the function approach
∞. And as the values of 𝑥 approach
zero from the negative direction, the outputs of the function approach negative
∞.
The 𝑦-axis is also a vertical
asymptote of the function. We want to translate the graph of
this function two units left and one unit down. And the easiest way to do this is
to first translate its asymptotes. Of course, vertically translating a
vertical line won’t change its position. Similarly, horizontally translating
a horizontal line won’t change its position. So we only need to translate the
vertical asymptote two units left and the horizontal asymptote one unit down. Translating the line 𝑥 equals zero
two units left gives us the line 𝑥 is equal to negative two and translating the
line 𝑦 is equal to zero one unit down gives us the line 𝑦 is equal to negative
one. These will be the asymptotes of our
function 𝑓 of 𝑥.
The general shape of the function
will remain unchanged because we’re only translating it horizontally and
vertically. It will still have the same shape
as the reciprocal curve. However, before we sketch this
curve, it can be useful to determine things like the 𝑦- and 𝑥-intercepts of the
curve to determine its orientation on the plane. We can find the 𝑦-intercept by
substituting 𝑥 is equal to zero into 𝑓 of 𝑥. We see that 𝑓 evaluated at zero is
one-half minus one, which we can evaluate as negative one-half. So our curve 𝑦 is equal to 𝑓 of
𝑥 must intercept the 𝑦-axis at negative one-half. In fact, this is enough to now
sketch our curve since we know its general shape and 𝑦-intercept. This gives us the following
sketch.
And it’s worth noting we could find
the coordinates of the 𝑥-intercept by setting 𝑓 of 𝑥 equal to zero and
solving. We would get that the value of 𝑥
is negative one. This sketch now helps us justify
why the horizontal asymptote and vertical asymptote of the function are the lines 𝑦
is equal to negative one and 𝑥 is equal to negative two, respectively. Since we’re just translating the
reciprocal curve, we’re just translating its asymptotes. So translating the reciprocal curve
one unit down will translate its horizontal asymptote one unit down. We will have the horizontal
asymptote 𝑦 is equal to negative one. Similarly, translating the
reciprocal curve two units left will translate its vertical asymptote two units left
onto the line 𝑥 is equal to negative two.
Therefore, we were able to sketch
the graph of the function 𝑓 of 𝑥 is equal to one divided by 𝑥 plus two minus one
and find its horizontal and vertical asymptotes. We showed its horizontal asymptote
was the line 𝑦 is equal to negative one and its vertical asymptote was the line 𝑥
is equal to negative two.
