Opening Down

A parabola that opens down has a negative a, often just a minus in front: $-{x}^{2}+2x+3$. Treat that minus as a = -1 and take it out of the two x terms: $-\left({x}^{2}-2x\right)+3$. Every sign inside flips, so the +2x becomes -2x.

Complete the square inside: ${x}^{2}-2x$ is ${\left(x-1\right)}^{2}-1$, so the expression is $-\left({\left(x-1\right)}^{2}-1\right)+3$. Expanding the minus flips the -1 to +1: $-{\left(x-1\right)}^{2}+1+3$, which is $-{\left(x-1\right)}^{2}+4$.

The vertex is (1, 4), and since a is -1 it is the highest point: 4 is the largest value the expression ever takes. Check it with the axis formula: -2 over 2 times -1 is 1, and putting 1 into the original gives -1 + 2 + 3, which is 4.