Question

Use the given information to answer the questions and interpret key features. Use any method of graphing or solving.

Answer 1

A quadratic function describes the relationship between the number of products x and the overall profits for a company.

The roots of the quadratic function are given as x = 0 and x = 28. We also know the graph's vertex is located at (14, -40).

The quadratic equation can be written in terms of its roots x1 and x2 as:

`f(x)=a(x-x_1)(x-x_2)`

Substituting the given values:

`\begin{gathered} f(x)=a(x-0)(x-28) \\ \\ f(x)=ax(x-28) \end{gathered}`

We can find the value of a by plugging in the coordinates of the vertex:

`f(14)=a\cdot14(14-28)=-40`

Solving for a:

`a=(-40)/(-196)=(10)/(49)`

Substituting into the equation:

`f(x)=(10)/(49)x(x-28)`

The graph of the function is given below:

The company actually loses money on their first few products, but once they hit 28 items, they break even again.

The worst-case scenario is that they produce 14 items, as they will have a profit of -40 dollars. The first root tells us the profit will be 0 when 0 products are sold.