Standard form to vertex form (ACT/SAT). Game on.

Complete the square: $y = (x^{2} - 10x + 25) - 25 + 21 = (x - 5)^{2} - 4\textsf{.}$ The constant $-4$ is the minimum value of $y\textsf{.}$

Complete the square: $y = (x^{2} + 4x + 4) - 4 - 12 = (x + 2)^{2} - 16\textsf{.}$

Factor out $2$ from the $x$ terms, then complete the square: $y = 2(x^{2} - 4x + 4 - 4) + 3 = 2((x - 2)^{2} - 4) + 3 = 2(x - 2)^{2} - 5\textsf{.}$ The vertex is $(2, -5)\textsf{.}$

Factor out $-1$ from the $x$ terms, then complete the square: $y = -(x^{2} - 6x + 9 - 9) - 5 = -((x - 3)^{2} - 9) - 5 = -(x - 3)^{2} + 4\textsf{.}$ The constant $4$ is the maximum value of $y\textsf{.}$

Half of $5$ is $\dfrac{5}{2}\textsf{,}$ and $\left(\dfrac{5}{2}\right)^{2} = \dfrac{25}{4}\textsf{.}$ Then $y = \left(x^{2} + 5x + \dfrac{25}{4}\right) - \dfrac{25}{4} + 4 = \left(x + \dfrac{5}{2}\right)^{2} - \dfrac{9}{4}\textsf{.}$

Factor out $3$ from the $x$ terms, then complete the square: $y = 3(x^{2} + 4x + 4 - 4) + 7 = 3((x + 2)^{2} - 4) + 7 = 3(x + 2)^{2} - 5\textsf{.}$ The vertex is $(-2, -5)\textsf{.}$

Complete the square: $y = (x^{2} - 2x + 1) - 1 - 8 = (x - 1)^{2} - 9\textsf{.}$ The constant $-9$ is the minimum value of $y\textsf{.}$

Complete the square: $y = (x^{2} + 12x + 36) - 36 + 20 = (x + 6)^{2} - 16\textsf{.}$