What a nice way to end the year! My Algebra II/Precalc group, coming off conic sections, were confronted with this. I asked them to create a table, and hand-graph. Lots of reasonably interesting discussion about getting more data points, then about the discontinuity and asymptote (first time they have run into a vertical one), and that horizontal asymptote. We digressed to talk about point discontinuities (eg, $y = \frac{x^2 + x}{x+1}$), and then I got a moment of satisfaction.

$y = \frac{1}{x} + 5$. Someone knew it was a vertical shift.

$y = \frac{1}{x+2}$. Someone guessed it was a horizontal shift. Verified.

$y = \frac{10}{x}$. Guess: vertical stretch. Verified.

Someone asked about a horizontal stretch. We looked at $y = \frac{1}{4x}$ and $y = (\frac{1}{4}) (\frac{1}{x})$ to get the relationship.

And then $y = \frac{4x + 9}{x+2}$ didn’t really bother anyone. Someone started dividing right away.

Transforming functions? Taught so much earlier? It sunk in.