SOLUTION: 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

Question 529599: 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.
(x = 1 is the day tickets go on sale).
Tickets = -0.6x2 + 12x + 11
Does the graph of the equation above open up or down?
How did you determine whether the graph of the equation above opened up or down?
Describe what happens to the ticket sales as time passes.
Use the quadratic formula to determine the last day that tickets will be sold.
Hint: Write your answer in terms of the number of days after ticket sales begin and you can�t have partial days.
Will tickets peak or be at a low during the middle of the sale?
How do you know?
After how many days will the peak or low occur?
Hint: Use the axis of symmetry formula.
How many tickets will be sold on the day when the peak or low occurs?
Hint: Substitute your value from question 7 into the original quadratic equation.
What are the coordinates of the vertex?
How did you determine this?
How many solutions are there to the original quadratic equation given in question 3?
How do you know?

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

The function above represents a parabola.
Unless you are studying Calculus, you are forced to rely on the parts of algebra that are the basis for calculus (often taught in a course named pre-calculus).
Parabolas are all derived from
which could be considered the mother of all parabolas and is a smile shaped curve extending up in both directions from a low point (minimum) in the middle.
A positive number other that 1 multiplying the just stretches the curve vertically or horizontally. A negative number flips the curve so it does not smile any more.
1) The graph opens down
2) How did you determine whether the graph of the equation above opened up or down? Because , and a negative number in front of the means it opens down, looks like a frowny mouth, and has a maximum.
3) Describe what happens to the ticket sales as time passes.
We know that eventually they will decrease. The equation predicts an increase, followed by a maximum, and then decrease. However, we have to figure out if the maximum comes at or it happened before opening day. We could try the completing the square trick, but a formula has been deduced for the generic quadratic function

that give us the x for the vertex (the axis of symmetry formula) as
and in this case

So, day 10 is the maximum for ticket sales. Tickets sold increase until then, and decrease after that.
4) Use the quadratic formula to determine the last day that tickets will be sold. The formula for the vertex position and the quadratic formula

both derive from the same complete-the-square algebra trick done on the generic quadratic function. In this case

So the function predicts that you would sell zero tickets on day 20.88, some positive number of tickets on day 20 (you can calculate that number), and negative number of tickets on day 21. Day 20 is the last day to expect any sales.
5) Will tickets peak or be at a low during the middle of the sale? They will peak. That was answered along with questions 1), 2), and 3).
6) How do you know? Because the function has a maximum, and it happens after day 1, as calculated above.
7) After how many days will the peak or low occur? On day 10, as calculated above.
Hint: Use the axis of symmetry formula. Thanks, but we needed that for question 3), didn't we?
8) How many tickets will be sold on the day when the peak or low occurs?
Hint: Substitute your value from question 7 into the original quadratic equation.
For

9) What are the coordinates of the vertex? They are (10,71),

as found above.
10) How did you determine this? We used the axis of symmetry formula to find and substituted in the tickets function to find .
11) How many solutions are there to the original quadratic equation given in question 3? Two solutions, approximated as -0.88 and 20.88.
12) How do you know? They were calculated in 4) above, but we know that the dicriminant determines if there is none, one, or two real solutions to the equation. The determinant is the part of the quadratic formula that is under the square root sign

If the discriminant is negative, there are no real number solutions. If it is zero, then is the only solution, and the parabola just touches the x-axis at that point. If the discriminant is positive, then we calculate its square root and find the two solutions, the points where the parabola crosses the x-axis, symmetrically placed at a distance of on both sides of the axis of symmetry.

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