Question

6.) the following function represents the profit P(n) in dollars that a concert promoter makes by selling tickets for n dollars each:
P(n)=-250n^2+3,250n-9,000

Part A: What are the zeroes of the above function and what do they represent? Show your work.

Part B: Find the maximum profit by completing the square of the function P(n ). Show the steps of your work.

Part C: What is the axis of symmetry of the function P(n )?

Answer 1

For Part A, what to do first is to equate the given equation to zero in order to find your x intercepts (zeroes)

0=-250n^2+3,250n-9,000 after factoring out, we get

-250(n-4)(n-9) and these are your zero values.

For Part B, you need to square the function from the general equation Ax^2+Bx+C=0. So to do that, we use the equated form of the equation 0=-250n^2+3,250n-9,000 and in order to have a positive value of 250n^2, we divide both sides by -1

250n^2-3,250n+9,000=0

to simplify, we divide it by 250 to get n^2-13n+36=0 or n^2-13n = -36 (this form is easier in order to complete the square, ax^2+bx=c)

in squaring, we need to apply (b/2)^2 to both sides where our b is -13 so,
(-13/2)^2 is 169/4

so the equation now becomes n^2-13n+169/4 = 25/4 or to simplify, we apply the concept of a perfect square binomial, so the equation turns out like this 

(n-13/2)^2 = 25/4 then to find the value of n, we apply the square root to both sides to obtain n-13/2 = 5/2 and n is 9. This gives us the confirmation from Part A. 

For Part C, since the function is a binomial so the graph is a parabola. The axis of symmetry would be x=5.