Question 188428: tickets= 0.2x^2+12x+11
A. Does this graph gp up or down? How did you determine this?
B. Describe what happens to ticket sales as time passes?
C. Use the quadrtic equation to determine the last day that tickets will be sold. (write your answer in terms of the number of days after ticket sales begin)
D. will tickets peak or be at a low during the middle of the sales? How do you know?
E. After how many days will the peak or low occur.
F. How many tickets will be sold on the day when the peak or low occurs?
G. What is the point of the vertex? How does this number relate to your answers in parts e anf f?
H. How many solutions are there in -0.2x^2+12x+11=0? How ndo you know?
I. What do the solutions represent? Is there a sloution that does not make sense? If so, in what ways does the solution not make sense?
Please help me solve these questions. I am not understanding this!!!''
Thank you for your help....
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You have a quadratic polynomial of the form:
If you want to graph this you need to say:
If a > 0, then the graph opens upward. If a < 0, then the graph opens downward. (if a = 0, you no longer have a quadratic, and you have a straight line for a graph. The quadratic polynomial that you represented at the beginning of your problem is one with a > 0, meaning that the graph opens upward.
This doesn't make sense for three reasons, a parabola graphing the value of something as a function of time and opening upward would model something that had a high value at time values near zero, would then decrease to a minimum, and then increase again. That sort of picture doesn't make sense for a ticket sales scenario. Secondly, looking at the graph of  (the green parabola below), you can see that ticket sales would start at 11 on day 0 and then be ever increasing. There is no way to determine the 'last day' from the model without knowing the maximum number of tickets that could be sold. Thirdly, you reprise the polynomial as a quadratic equation in part H of the question, and here you have a negative sign on the lead coefficient. Hence, I suspect (and will assume going forward) that you simply left the minus sign off when you typed in the trinomial expression.
B. Given the assumption above, the ticket sales will increase from 11 on day 0 (presumeably, a few were sold prior to the start of sales). You know this because if you substitute 0 for x, y = 11. Reaching a peak somewhere around day 30 or so, then falling off to zero about day 60ish (we'll nail this down in the next part).
C. In order to use the quadratic formula, you need to have a quadratic equation, so you must set your quadratic polynomial equal to zero:
then:
(11)}}{2(-0.2)})
I'll let you do your own arithmetic, but will tell you that you will, of course, get two roots to the equation. One of them will be a smallish negative number that makes no sense because the last day of ticket sales certainly can't occur before you start selling tickets. So the other answer is the one you want. Remember though, even though the answer you get will be an irrational number represented by a decimal fraction, you need to report the number of days as an integer -- but remember to report it this way: Since the time between day 0 and day 1 is the first day, the time between day 5 and 6 is the sixth day, and so on, the right answer to your question will be the next higher integer from the actual root you determined.
D. A peak. Look at the graph.
E, F, and G. The vertex of the parabola represented by the equation:
 = y = ax^2 + bx + c)
has an x-coordinate of 
and a y-coordinate of the value of the function at the x-coordinate of the vertex.
 = f\left( -\frac{b}{2a}\right) = a \left( -\frac{b}{2a}\right)^2 + b\left( -\frac{b}{2a}\right) + c)
Part G asks for the coordinates of the vertex, so:
}=30)
 = -0.2(30)^2 + 12(30) + 11 = -180+360+11=191)
Meaning that the vertex is at (30, 191). The x-coordinate of the vertex is also the day, 30, when the most tickets will be sold in answer to part E. And the y-coordinate of the vertex, 191, is number of tickets that will be sold that day in answer to part F.
H and I. Actually, you can get the answers to this from part C. You saw that there are two solutions. In fact there are always two solutions for any quadratic. Ok, when you have a perfect square trinomial, you have one solution, but it has a multiplicity of 2. We know this because the Fundamental Theorem of Algebra says that for any polynomial of degree n there are n factors of the form (x - a), and a quadratic equation is of degree 2, so there must be 2 factors of the polynomial, hence 2 roots of the equation. Sometimes the roots are complex numbers (a + bi), but there are always 2. As to this problem, we already discussed, in part C, that only one of the roots makes sense in terms of the story behind the problem.
John
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