Question 628162: tickets= - 0.4x^2 + 13x + 14
use the quadratic equation to determine the last day tickets will be sold and how many will be sold on the day the peak or low occurs and write the answers in terms of the number of days after ticket sales begin. What is the point of the vertex? How does this number relate to your answers
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! I'll help as well as I can but I suspect that there might be something incorrect in the equation you posted:- x = 0 represents the start of ticket sales. But if you put zero in for x into your equation, you get 14 tickets sold. How does one sell 14 tickets before the start of ticket sales?
- Usually the low of tickets sales is on the day that sales drop to zero. And usually this day works out to be a whole number. But with your equation, the day works out to be a mixed number.
Neither of these mean that there must be an error in the equation but they make me suspicious. So I will tell you what I can.
First let's picture what the graph of this equation looks like. An equation of two unknowns, x (days) and tickets, where just one of them is squared is the equation of a parabola. And when the coefficient of the squared term is negative, this parabola will open downward (like an upside-down U). Picture a parabola like this in your mind (or draw one on a piece of paper). You should be able to see that the vertex of the parabola will be the highest point of this parabola. So the coordinates of the vertex represent the day and the number of tickets sold on that day for the day when the most tickets are sold. To find the vertex we can either- Complete the square and put the equation into vertex form:
 ; or - Remember that when the equation is in the
 form the x-coordinate of the vertex is -b/2a
Your equation is already in the second form so we'll use that way to find the vertex. The x-coordinate of the vertex of your equation is:
As you can see, we get a mixed number. The only way to interpret this is to assume that the equation is an imperfect model of the situation and so we must take the answer it gives us with "a grain of salt". In other words, round off. So we read 16.25 as meaning that the maximum ticket sales will be on the 16th day of sales. To find the number of tickets sold on that day, just put 16.25 in the equation for the x's and simplify. You will again get a mixed number. Since fractions of tickets are not sold, just round off to the nearest whole number.
At some point this parabola will come back down and hit the x-axis. This represents when zero tickets are sold. Obviously zero tickets sold would be the lowest number sold (since you can't sell a negative number of tickets).To find out when this happens, just set the number of tickets to zero and solve for x:
To solve this just use the Quadratic Formula. It will give you two solutions. One of them will be a negative number. Reject/discard/ignore this one because a negative value for x represents a day before the start of ticket sales. The positive solution will be another mixed number. Round this off to the nearest whole number. This will be the day of the lowest ticket sales.
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