A. I would suggest including all units in your answer. Both answers are correct but the unit is also important.
average rate of change:
($750 - $350) / (2014 - 2010) = $400 / (4 years) = $100 per year
initial value:
$350
B. Take 2010 to be year 0. Then for year x = 0, the fee is y = $350.
If the fee increases at a rate of $100 per year, this means that when x = 1, y = $350 + $100 = $450. So the graph of the function that models the dog-walking fee is a line that passes through the points (0, 350) and (1, 450), adn the slope of this line is
(450 - 350) / (1 - 0) = 100/1 = 100
which is the same as the average rate of change.
Using the point-slope formula for a line, the equation you want is then
y - 350 = 100 (x - 0)
y - 350 = 100x
y = 100x + 350
