Expert: Scott A Wilson - 5/27/2011

Question
QUESTION: with the formula: tickets = -.2x^2 + 12x +11
does the graph open up or down?

ANSWER: The sign of the x^2 term determines this.
If it were positive, it would open up.
Since it is negative, it opens down.


---------- FOLLOW-UP ----------

QUESTION: 1.   Suppose you are an event coordinator for a large performance theater. One of the hottest new Broadway musicals has started to tour and your city is the first stop on the tour. You need to supply information about projected ticket sales to the box office manager. The box office manager uses this information to anticipate staffing needs until the tickets sell out. You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. (  is the day tickets go on sale).   
         

  a. Does the graph of this equation open up or down? How did you determine this?
  
b. Describe what happens to the tickets sales as time passes.

c. Use the quadratic equation to determine the last day that tickets will be sold.

Note. 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 sale? 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. and f?
  
h. How many solutions are there to the equation ? How do you know?
  
i. What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense?

Answer
I put the equation in Excel, then copied in the results:

0   11
1   22.8
2   34.2
3   45.2
4   55.8
5   66
6   75.8
7   85.2
8   94.2
9   102.8
10   111
11   118.8
12   126.2
13   133.2
14   139.8
15   146
16   151.8
17   157.2
18   162.2
19   166.8
20   171
21   174.8
22   178.2
23   181.2
24   183.8
25   186
26   187.8
27   189.2
28   190.2
29   190.8
30   191
31   190.8
32   190.2
33   189.2
34   187.8
35   186
36   183.8
37   181.2
38   178.2
39   174.8
40   171
41   166.8
42   162.2
43   157.2
44   151.8
45   146
46   139.8
47   133.2
48   126.2
49   118.8
50   111
51   102.8
52   94.2
53   85.2
54   75.8
55   66
56   55.8
57   45.2
58   34.2
59   22.8
60   11
61   -1.2

By looking at this, it can be seen to have a max at t=30,
then decrease on both sides (opens down).
Since it opens down, as time passes, the change in ticket sales decreases.
Solve settting the equation equal to zero to determine when sales will end.

Question d can be answered by looking at the curve.
The peak can be seen on the graph to be at 30 days, right?
The low will be when sales are 0, and
Putting x=60 in is the last day a positive value is generated.
There will be 191 at the peak.
The vertex is where the maximum sales occur.

Since it is a quadratic, there are two solutions, though one of them is negative.
This can be thought of as being a few days before sales opened.
This is the solution that doesn't make sense.