I need to complete some important questions for math, but I am unsure of how to complete them.

Question

Here are the questions I need help with: 2. The parabola has the roots of ± √13 and passes through the point (-5, -4). Determine the equation of the parabola. Identify the vertex of the quadratic function. 

3. Determine max/min value and when it happens using the indicated method.
(complete the square)
4. For what value(s) of k will the function f(x) = kx^2 -4x +k have:
(a) two equal real roots?                         (b) no zeros?
5. Visakan is making an arch at the top of his barn doors. The arch has a shape that can be modelled by the equation of a parabola. If the left edge of the door is the origin and the doorway is 6m wide, what is the equation of the parabola if the height of the arch is 2m from the edge of the doorway is 5m?

6. Stephen is throwing a horseshoe at a wooden stake. The function h(t) = -5t^2 +30t +1 models the height of the horseshoe in meters above the ground at time t. A tree 10m tall is between Stephen and the wooden stake.(a) What is the maximum height of the horseshoe? What is the minimum height?
(b) Will the horseshoe hit the tree? Explain your answer.
(c) About how long will it take for the horseshoe to reach the ground?

7. The demand function for a new perfume is p(x) = -2x +36 where p(x) represents the selling price, in thousands of dollars, and x is the number of bottles sold, in thousands. 
a. Determine the revenue function and maximum revenue
b. At what price will the company receive maximum profits? 

Answer 1

#2. y = a(x^2-13)
using the point, a(25-13) = -4, so a = -1/3
y = -1/3 (x^2-13)
so the vertex is at (0, 13/3)

#3. for y=ax^2+bx+c the vertex (max/min) is at x = -b/2a

#4. kx^2 - 4x + k has discriminant d=16-4k^2
two equal roots, d=0
no real roots, d<0

#5. the parabola is y=a(x-3)^2 + b
and we need y(2) = 5
Unfortunately, there are many parabolas that fit this, since you do not say how tall the door is.

#6. max height is at t = -b/2a = 3
so see whether that is at least 10.
for flight time, find when h=0

#7. revenue is price * quantity
now find the vertex of that parabola, as done above