An instructor gives an exam with sixteen questions. students are allowed to choose any eleven to answer.

(a) how many different choices of eleven questions are there? 1365 correct: your answer is correct.
(b) suppose six questions require proof and nine do not.
(i) how many groups of eleven questions contain four that require proof and seven that do not? 540 correct: your answer is correct.
(ii) how many groups of eleven questions contain at least one that requires proof? 1365 correct: your answer is correct. (iii) how many groups of eleven questions contain at most three that require proof? 176 incorrect: your answer is incorrect.
(c) suppose the exam instructions specify that at most one of questions 1 and 2 may be included among the eleven. how many different choices of eleven questions are there?
(d) suppose the exam instructions specify that either both questions 1 and 2 are to be included among the eleven or neither is to be included. how many different choices of eleven questions are there?

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

step-by-step explanation:

if x decreases by 4 then y increases by 10

step-by-step explanation:

i dont know

step-by-step explanation:

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1