Pearson Algebra 2 Common Core, 2011
PA
Pearson Algebra 2 Common Core, 2011 View details
2. The Reciprocal Function Family
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Exercise 31 Page 513

Let's start by reviewing the parent function This function has two branches, and its lie on the and This means that its vertical asymptote is and its horizontal asymptote is

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, is a scale factor of 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 by replacing the independent variable by
If is positive, this will translate the original function units to the right. If is negative, the original function will be translated 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
We can now take a look at its graph.

As we can see, since the graph of the function, as well as the vertical asymptote, was translated units to the right — as expected. The equation for the vertical asymptote is now while the horizontal asymptote is still 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 by adding the constant
If is positive, this will translate the original function units upwards. And if is negative, the original function will be translated units downwards. Note that this will translate the horizontal asymptote of the original function in the same way. Let's see an example using
We can now take a look at its graph.

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