Question 626969
This is a very long question I will now answer only the first part. 

a. The equation of the parabola is {{{y = -2x^2/5 +13x+14}}}. 

Reorganizing the equation
{{{y = -2x^2/5 +13x+14}}}
{{{y = (-2/5)(x^2 - 65x/2 - 35)}}}
{{{y = (-2/5)((x - sqrt(65)/4)^2 - 65/16 - 35)}}}

As the coefficient of {{{x^2}}} is negative so the parabola is facing downwards.


You can verify this from the graph below.
{{{graph( 300, 200, -10, 40, -20, 180, -0.4*x^2 + 13*x + 14 )}}}