### Video Transcript

The graph shows π¦ equals π over
π₯ minus π plus π. A single point is marked on the
graph. What are the values of the
constants π, π, and π?

We begin by noticing that this
graph resembles the graph of π¦ equals one over π₯. We can obtain the given graph from
the graph of the parent function π¦ equals one over π₯ by applying some function
transformations. The graph of the parent function π¦
equals one over π₯ has a horizontal asymptote at π¦ equals zero and a vertical
asymptote at π₯ equals zero. The given graph has a horizontal
asymptote at π¦ equals negative two and a vertical asymptote at π₯ equals three. This means that a downward shift of
two units and a rightward shift of three units is one of the function
transformations used to obtain this graph from the graph of π¦ equals one over
π₯.

Before applying this translation,
however, we first need to check if any other transformations are involved, since the
order of transformations is very important. There are three different types of
transformations to consider: translation, dilation, and reflection. The given graph is oriented the
same way as the parent function, so we can rule out reflection. Dilation is a possibility, but itβs
difficult to judge by eye. Since we are given the point six,
negative one on the graph, we can use this point to determine the dilation factor,
if there is one. Recall that when combining
transformations, dilations and reflections must be done before translations.

Recall that a horizontal dilation
by a scale factor of π one means that we map π₯ to π₯ over π one. In other words, the π₯-values of
all the points on the graph are reduced by a factor of π one. A vertical dilation by a scale
factor of π two means mapping π of π₯ to π two times π of π₯. In other words, we multiply all of
the π¦-values on the graph by a factor of π two. Beginning with the parent function
π of π₯ equals one over π₯ and performing a horizontal dilation, we get one over π₯
over π one.

Next, performing a vertical
dilation, we multiply this new π of π₯ by π two, giving π two over π₯ over π
one. This simplifies to π one π two
over π₯. π one times π two is just another
constant in and of itself. So, we can call this constant
π. We do not yet know the value of
π. First, we need to move on to
translations. The positions of asymptotes are
unaffected by dilations. Therefore, these could only have
been moved by the translation of two units downwards and three units to the
right. A vertical translation of π units
means mapping π of π₯ to π of π₯ plus π. So, we change all of the π¦-values
of the points on the graph from π¦ to π¦ plus π. π being positive means the graph
moves upwards and negative means it moves downwards.

In our case, we need to perform the
dilation first. So, the parent function one over π₯
goes to π over π₯. Then, to translate this by two
units downwards, this goes to π over π₯ minus two. Now, recall that for a horizontal
translation by π units, we map all of the π₯-values to π₯ minus π. Again, the value of π will
determine the direction, with positive π giving a rightward shift and negative π
giving a leftward shift. But of course, the sign will be
reversed since we are subtracting π from π₯. So, starting from π over π₯ minus
two, to perform a rightward shift of three units, we need to change the π₯-values to
π₯ minus three. So, this gives us π over π₯ minus
three minus two. So, this graph represents the
function π of π₯ equals π over π₯ minus three minus two. So, we have found two of the
parameters in the question. π is equal to three and π is
equal to negative two.

Now we just need to find the value
of π. This equation has three unknowns in
it. We have π of π₯, π₯, and π. We are given a point on the graph:
six, negative one. And this corresponds to values of
π of π₯ and π₯, respectively, which we can then substitute into the equation and
rearrange to find π. So, we have π of π₯ equals
negative one and π₯ equals six. Simplifying and rearranging gives
π equals three. So, we now have the values of all
three constants. π is equal to three, π is equal
to negative two, and π is equal to three.