Let's start by reviewing the parent function $y = \frac{1}{x} .$ This function has two branches, and its $asymptotes$ lie on the $x -$ and $y - axes .$ 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 .

y_{1} = \frac{2}{x} \to y_{2} = \frac{2}{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 .

y_{1} = \frac{2}{x} \to y_{2} = \frac{2}{x - 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 .

y_{1} = \frac{2}{x} \to y_{3} = \frac{2}{x} + 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 .

y_{1} = \frac{2}{x} \to y_{3} = \frac{2}{x} + 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.

Exercise