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:
[tex]f(x)=a(x-x_1)(x-x_2)[/tex]
Substituting the given values:
[tex]\begin{gathered} f(x)=a(x-0)(x-28) \\ \\ f(x)=ax(x-28) \end{gathered}[/tex]
We can find the value of a by plugging in the coordinates of the vertex:
[tex]f(14)=a\cdot14(14-28)=-40[/tex]
Solving for a:
[tex]a=\frac{-40}{-196}=\frac{10}{49}[/tex]
Substituting into the equation:
[tex]f(x)=\frac{10}{49}x(x-28)[/tex]
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.
