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Let's start by reviewing the parent function y=x1. 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, y1=x2, 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.
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.
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.