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= -(2)^2 + 4(2) + m
= -4 + 8 + m
m = 8
The value of $ m $ is $ \boxed{8} $.
Question: A science communicator models the growth of a YouTube channel's subscribers with $ S(t) = at^2 + bt + c $. If $ S(1) = 500 $, $ S(2) = 1200 $, and $ S(3) = 2100 $, find $ a $.
Solution: Set up equations using the given points:
For $ t = 1 $: $ a(1)^2 + b(1) + c = 500 \Rightarrow a + b + c = 500 $ (1)
For $ t = 2 $: $ 4a + 2b + c = 1200 $ (2)
For $ t = 3 $: $ 9a + 3b + c = 2100 $ (3)
Subtract (1) from (2): $ 3a + b = 700 $ (4)