If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4$?
the maximum value occurs when...
t = -8 / [ 2(-1) ]
t = -8 / -2
t = 4
When t = 4 ...
-t2 + 8t - 4
= -42 + 8(4) - 4
= -16 + 32 - 4
= 12
Here's a graph: https://www.desmos.com/calculator/qdghoci3o9
the maximum value occurs when...
t = -8 / [ 2(-1) ]
t = -8 / -2
t = 4
When t = 4 ...
-t2 + 8t - 4
= -42 + 8(4) - 4
= -16 + 32 - 4
= 12
Here's a graph: https://www.desmos.com/calculator/qdghoci3o9