The function f(x) = −x2 + 44x − 384 models the daily profit, in dollars, a shop makes for selling donut combos, where x is the number of combos sold, 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) (10 points)

Let's first set this equal to 0 so we can factor it.  We will factor by completing the square, because completing the square gets this into vertex form, which is what we want.  Then we can determine what the vertex actually means to us in our particular situation.  To save steps we will do a couple at a time, if possible.  We will set the polynomial equal to 0, then at the same time, move the constant over to the other side:  .  The rules for completing the square are very specific.  We have to have a 1 as our leading coefficient in order to complete the square properly.  Ours is a -1, so we have to factor it out:  .  Now we can complete the square.  Take half the linear term, square it and add that number to both sides.  Our linear term is 44.  Half of 44 is 22, and 22 squared is 484.  We add 484 into the parenthesis, but we cannot forget about the -1 hanging around out front there.  It is a multiplier that cannot be disregarded.  What we have actually multiplied in is -1(484) which is -484.  Add that to the right now.  .  What we have done in this process is created a perfect square binomial on the left.  .  If we add 100 to both sides we can find the vertex.  .  The vertex is (22, 100).  The x coordinate of the vertex is the number of donut combos sold and the y coordinate is the amount of money made by selling that many combos.  Because this is an upside down parabola, our vertex is a max value.  That means that our profits will be the highest at this point.  By selling 22 combos, our max profit is $100.  Selling even one more or one less gives us a lower profit.  That's part A.  For part B, the x-intercepts occur when y = 0.  That means that when we find the x values where y is 0, we have made that many donuts and make no profit.  We will find the x-intercepts by factoring.  These x-intercepts are also called the roots, the solutions, and the zeros of the function.  When we factor the original polynomial, we get zeros of x = 12 and x = 32.  That means that if we sell 12 combos we make no money and if we sell 32 combos we make no money.  Why this is all depends upon the company's cost of actually making the donuts.  We don't know that.  All we know is that when x = 12 combos, the money is 0 and when x = 32 combos, the money is 0.