This is a concept check on quadratic equation application. I am having trouble with a couple of parts. If anyone can help I would appreciate it.

Suppose you are an event coordinator for a large performance theater. You need to supply information about projected ticket sales to the manager. You provide a quadratic equation that models expected number of sales for each day 'x'. (x=1 is the day tickets go on sale."
Tickets=-2x^2+12x+11

I answered a.
B. Describe what happens to the ticket sales as time passes.

I am not sure how to determine this. Do I determine this by the vertex or line of symmetry?

I answered C.
D.Will tickets peak or be at a low during the middle of the sale? How do you know?

Again, this question stumps me. How do you tell something like this with a parabola?

I would appreciate a detailed explanation. I don't want answers as much as how to determine the information.

Thanks for helping.

B. The direction of the parabola, which is downwards because of the -2x^2 term, will tell you that the ticket sales will decrease as x increases.

e.g. if x = 2 , Sales = -8 + 24 + 11 = 27
if x = 3, sales = -18 + 36 + 11 = 29
if x = 4, sales = -32 + 48 + 11 = 27
if x = 5 , sales = -50 + 60+11 = 21 , notice the sales are decreasing
it also looks like the vertex is at (3,29)
( Assume you know how to find the vertex)

so as x gets bigger (time passes), the first term becomes more negative at a faster rate than the two positive terms, so the sum (the ticket sales) gets smaller, until it becomes meaningless when ticket sales is a negative number.
(e.g. x = 10, sales = -200 + 120 + 11 = -69 )

I suggest you sketch the relation after making a table of values for x = 1,2,3,4,5,6,7

However keep in mind that you don't have a continuous curve, but rather a series of integer valued points, which will fall along the parabola
The vertex of (3,29) tells you that there was a maximum number of ticket sales of 29 on day 3
notice when x=6, sales = -72 + 72 + 11 = 11
but when x-7, the graph no longer has any real meaning.

Thanks a bunch. I appreciate the detail. quadratic equations are confusing for me to graph. I no sooner think I have it than I get an answer wrong.

To answer question B, "Describe what happens to the ticket sales as time passes," we need to analyze the quadratic equation. The equation you provided is Tickets = -2x^2 + 12x + 11. This equation represents a parabola.

The graph of a quadratic equation is a U-shaped curve called a parabola. In this case, the parabola opens downward because the coefficient of the x^2 term (-2) is negative. The x-axis represents time, while the y-axis represents the number of ticket sales.

To determine what happens to ticket sales as time passes, we can look at a few key points on the graph of the parabola:

1. Vertex: The vertex is the highest or lowest point on the parabola. It represents the peak or lowest point of ticket sales. In this case, the vertex is given by the formula x = -b/2a, where a = -2 (coefficient of x^2 term) and b = 12 (coefficient of x term). Plugging these values into the equation, we get x = -12 / (2 * -2) = 12/4 = 3. Therefore, the vertex is at x = 3.

2. Line of Symmetry: The line of symmetry is a vertical line that divides the parabola into two equal halves. The x-coordinate of the vertex is the equation of the line of symmetry. In this case, the line of symmetry is x = 3.

Now let's apply this information to your questions:

Question B: Describe what happens to the ticket sales as time passes.
The parabola opens downward, and the vertex is at x = 3. As time passes (i.e., as x increases), the ticket sales initially increase until they reach the peak at x = 3 and then start decreasing.

So, in simple terms, ticket sales will increase and then decrease as time passes. The peak number of ticket sales will occur when x = 3, and after that, the ticket sales will decline.

Moving on to question D: Will tickets peak or be at a low during the middle of the sale? How do you know?
Since the vertex represents the highest or lowest point of the parabola, we can conclude that ticket sales will peak at x = 3, which is the middle of the sale. This is because the parabola is symmetric with respect to its vertex, and the highest or lowest point occurs in the middle.

To summarize, by analyzing the vertex and the graph of the quadratic equation, we can determine that ticket sales will increase and then decrease as time passes. The peak number of ticket sales will occur at x = 3, the middle of the sale.