Question

the price is right
the state fair is in town! the fair is selling ticket packages for the rides it offers. the package includes passes for the ferris wheel, roller coaster, bumper cars, and merry - go - round. the ticket sales can be modeled by the function ( p(x)=-25x^{2}+525x ), where ( x ) is the price of the ticket. without creating graphs, answer the following.

1. rewrite ( p(x) ) in its equivalent vertex form. what is the vertex of this function and what does in mean in relation to ticket sales?
2. using vertex form, write an equation that can be used to determine the ticket price that gives a revenue of $0.
3. solve the equation you wrote in #2. show or explain your work.
4. using vertex form, write an equation that can be used to determine the approximate ticket price that gives revenue of $1200.

Explanation:

Step1: Recall the vertex form of a quadratic function

The vertex form of a quadratic function is \(y = a(x - h)^2 + k\), where \((h,k)\) is the vertex of the parabola. For the function \(p(x)=-25(x - 10.5)^2+2756.25\), by comparing with the vertex form \(y = a(x - h)^2 + k\), we can directly identify the values of \(h\) and \(k\).

Step2: Identify the vertex coordinates

In the function \(p(x)=-25(x - 10.5)^2+2756.25\), \(h = 10.5\) and \(k = 2756.25\). So the vertex of the function \(p(x)\) is \((10.5,2756.25)\). The \(x\) - coordinate of the vertex represents the price of the ticket (\(x\) is the price of the ticket in the context of the problem), and the \(y\) - coordinate represents the revenue.

Answer:

The vertex of the function \(p(x)\) is \((10.5,2756.25)\). The vertex \((10.5,2756.25)\) means that when the ticket price \(x = 10.5\) dollars, the revenue \(p(x)=2756.25\) dollars.