Exercise 31 Page 513

Let's start by reviewing the parent function $y=\frac{1}{x}.$ This function has two branches, and its $\text{asymptotes}$ lie on the $x\text{-}$ and $y\text{-axes}.$ This means that its vertical asymptote is $x=0,$ and its horizontal asymptote is $y=0.$

We will be labeling each function from now on to distinguish between them. Notice that the only difference between the parent function and the given function, $y_1=\frac{2}{x},$ is a scale factor of $2.$ This means that it is stretched, but the asymptotes remain the same.

Now, we can discuss how to find an example for the required horizontal translation and for a vertical translation being applied to the given function. We will discuss each case individually.

Horizontal Translation

We can apply a horizontal translation to the function $y_1=\frac{2}{x}$ by replacing the independent variable $x$ by $x-h.$ If $h$ is positive, this will translate the original function $h$ units to the right. If $h$ is negative, the original function will be translated $|h|$ units to the left. Note that this will translate the vertical asymptote of the original function in the same way. Let's see an example using $h=2.$ We can now take a look at its graph.

As we can see, since $h=2$ the graph of the function, as well as the vertical asymptote, was translated $2$ units to the right — as expected. The equation for the vertical asymptote is now $x=2,$ while the horizontal asymptote is still $y=0.$ Note that there are infinitely many examples of horizontal translations, this is just one of them.

Vertical Translation

We can apply a vertical translation to the function $y_1=\frac{2}{x}$ by adding the constant $k.$ If $k$ is positive, this will translate the original function $k$ units upwards. And if $h$ is negative, the original function will be translated $|h|$ units downwards. Note that this will translate the horizontal asymptote of the original function in the same way. Let's see an example using $k=3.$ We can now take a look at its graph.

As we can see, since $k=3,$ the graph of the function, as well as the horizontal asymptote, was translated $3$ units upwards — as expected. The equation for the horizontal asymptote is now $y=3,$ while the vertical asymptote is still $x=0.$ Note that there are infinitely many examples of vertical translations, and this is just one of them.