Both at Once

The two kinds of move combine. Take y = f(x + 1) - 2 with f(x) = x^2. Inside, x + 1 moves the graph 1 left. Outside, the minus 2 moves it 2 down. So the vertex goes from (0, 0) to (-1, -2).

The pattern is worth naming: for f(x) = x^2, y = f(x - h) + k has its vertex at (h, k). Read h from inside the brackets, where the sign looks flipped (x + 1 is x - (-1), so h is -1), and read k from outside, exactly as written. With f(x) = x^2 this is y = (x - h)^2 + k, called the vertex form of a parabola.

The order of the two moves does not matter: slide then lift, or lift then slide, you land in the same place. After the vertex, the shape does the rest. One step right of it the curve is 1 higher, so (0, -1) is on this graph too.