Living in or near a metropolitan area has some advantages. Entertainment opportunities are almost endless in a major city. Events occur almost every night, from sporting events to the symphony. Tickets to these events are not available long and can often be modeled by quadratic equations.
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?
a. To determine whether the graph of the quadratic equation opens up or down, we need to examine the coefficient of the quadratic term. If the coefficient is positive, the graph will open up, and if it's negative, the graph will open down.
b. As time passes, ticket sales will initially increase, reach a peak, and then start to decline.
c. To determine the last day that tickets will be sold, we need to find the point where the ticket sales reach zero. This can be done by setting the quadratic equation equal to zero and solving for the value of x, which represents the number of days after ticket sales begin.
d. Tickets will be at a peak during the middle of the sale. This is because the quadratic equation represents a symmetrical curve, with the peak occurring at the vertex of the graph.
e. To find the number of days when the peak or low occurs, we need to find the x-coordinate of the vertex.
f. The number of tickets sold on the day of the peak or low can be found by substituting the x-coordinate of the vertex into the quadratic equation.
g. The point of the vertex represents the maximum or minimum point on the graph of the quadratic equation. The y-coordinate of the vertex tells us the maximum or minimum value of the ticket sales, and how it relates to the answers in parts e and f.
h. The number of solutions to the equation depends on the discriminant of the equation. The discriminant is the value under the square root sign in the quadratic formula. If the discriminant is positive, there are two real solutions; if it's zero, there is one real solution; and if it's negative, there are no real solutions.
i. The solutions to the equation represent the number of days when ticket sales will be zero or when the event is sold out. If there is a solution that does not make sense, it would be an imaginary or complex solution, indicating that there is no real value for the number of days when ticket sales will be zero.