the function f(x) = –x2 + 60x – 116 models the monthly profit, in dollars, a shop makes for repairing windshields, where x is the number of windshields repaired, and f(x) is the amount of profit.

part a: determine the vertex. what does this calculation mean in the context of the problem? (5 points)

part b: determine the x-intercepts. what do these values mean in the context of the problem? (5 points)

Part A: If you know how to convert a quadratic to vertex form, you can tell the vertex just by looking at it. In the standard form it is a bit more complicated, but very possible. It will also help for part B. To find the vertex of a quadratic in this form you need the zeroes, or when the equation equals zero, and then find the center point of those two and plug that point into the equation. This is easily done with the quadratic formula, but if you cannot then facoring is the way to go.

Basically we're turning the quadratic into the form (x+a)(x+b) because if you foil that out you get x^2+(a+b)x+ab, which is identical to the starting form, with as and bs instead of numbers. To do this we need to do two things. The first, and easiest is making sure there is a 1 or -1 in the second degree term (the part that has the variable squared). That's already done so we move on. If we lebel each term as a, b and c so a is the first b is the second and c is the third we only want to use the numbers from each so -1, 60 and -116. For any quadratic, what you are trying to do is split the b term into two terms, so they will have to add up to 60. Keep in mind that this can mean something like -60+120, it does not in this case but others could. These two terms that add up to 60 also have to multiply together to equal whatever a*c equals. In this case -1*-116=116

I like to start with the factors of the multiplied term, 116, and see if I can get any to work. So starting with 1, 1*116=116 and -1*-116=116. Neither of these add together to make 60 though. SO next, since the number is even, we'll try 2. 2*58=116 and -2*-58=116. 2+58 also equals 60, so we are lucky enough that it is our second try. So now we have -x^2+2x+58x-116. If you have experience factoring by grouping you can take it from here, otherwise we want to separate this into two terms (-x^2+2x)+(58x-116) and the goal is to facor something out from each term in parenthesis so that they are equal.

The first term can have x or -x pulled out, I am going to pull -x out just for looks. Now we have -x(x-2) for that first term, it is still part of the whole problem though. The second can have 58 factored out so we get 58(x-2) and we see both terms in the parenthesis are equal, so now our equation looks like -x(x-2)+58(x-2). It may be hard to see, but we can factor the parenthesis out of each term of the whole equation. To better see it if we pretend (x-2) is just y we have -xy+58y. And you should be able to see we can factor the ys out and get y(-x+58) so if we stop pretending y=(x-2) then we have (x-2)(-x+58) and it is in the form we wanted. I will assume you can tell the zeroes from here.

The zeroes of course are the x intercepts, and then the x coordinate between the zeroes is the x coordinate of the vertex. Keep an eye out for quadratics without zeroes though. As for the context of each, think of it like this. Whatever number you plug in for x is always referring to how many windshields are repaired and f(x) is the profit. So if 0 are repaired there is a profit or -166, which is a loss, which makes sense. Also try and think of what f(x)=0, or the highest point possible you can get at f(x) means. For instance at f(x)=1 that means one dollar was earned. And since the lowest number you can repair is 0, the lowest amount you can make is -166 dollars by not repairing any windshields

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2 pecks

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