Reading the Rule

Once you know the moves, you can read a parabola straight from its rule, with no f in sight. Every rule $y = a{\left(x-h\right)}^{2}+k$ 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 $y = 2{\left(x-3\right)}^{2}-5$. 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 $y = 3{\left(x+2\right)}^{2}+1$, $x+2$ is $x-\left(-2\right)$, so h is -2 and the vertex is (-2, 1). In $y = -3{\left(x-1\right)}^{2}+2$, a is -3: the curve opens down from its top point (1, 2) and drops 3 one step away.