How do I find the rule?
level decreases at a constant rate ... H(n) is a linear function.
take two values for (n, H(n)) ...
(0, 86) (3, 68)
determine the slope ...
$\displaystyle \frac{\Delta H}{\Delta n} = \frac{68 - 86}{3 - 0} = -6$
use the point-slope form for a linear equation ...
H(n) - 86 = -6(n - 0)
H(n) = 86 - 6n
Is one of those points the vertex? Aren't those 3 points in a straight line; like I can't get the equations of two linear functions and find the vertex by finding their point of intersection if they're straight. Sorry if I've missed something, I'm a bit tired from studying all day.
having to find an absolute value function, I would sketch a graph with points at (0,86) (3,68) (6,50) (9,68) (12,86).
connecting the points using lines, one can see the large "V" indicating a transformed version of y = |x|.
slope is $\displaystyle \pm$ 6 ...
y = 6|x|
horizontal shift to the right 6 units ...
y = 6|x-6|
vertical shift up 50 units ...
y = 6|x-6| + 50
but, again, the problem statement should have been more clear in relating what was required as a solution.