Reading the Rule

Once you know the moves, you can read a parabola straight from its rule, with no f in sight. Every rule y = a(x - h)^2 + k with a not zero is the parent x^2 with every height multiplied by a, then moved so the vertex sits at (h, k).

Take y = 2(x - 3)^2 - 5. The bracket x - 3 is 0 at x = 3, and the minus 5 is outside, so the vertex is (3, -5). The 2 in front is a: one step from the vertex the curve rises 2, so it passes through (4, -3). Since a is positive, the curve opens up.

Watch the signs. In y = 3(x + 2)^2 + 1, x + 2 is x - (-2), so h is -2 and the vertex is (-2, 1). In y = -3(x - 1)^2 + 2, a is -3: the curve opens down from its top point (1, 2) and drops 3 one step away.