Question

1. the amount of profit bill makes per toy when he increases or decreases the price of his han be modeled by the function ( f(x)=-x^{2}-2x + 3 ). what price change gives him the highest profit per toy? (8pts) show work!!! price change: $ maximum profit per toy: $

Explanation:

Step1: Find the vertex of the quadratic function

The function \(f(x)=-x^{2}-2x + 3\) is a quadratic function in the form \(y = ax^{2}+bx + c\), where \(a=-1\), \(b = - 2\), \(c = 3\).
The \(x\) - coordinate of the vertex of a quadratic function \(y=ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\).
Substitute \(a=-1\) and \(b=-2\) into the formula:
\(x=-\frac{-2}{2\times(-1)}=\frac{2}{-2}=-1\)

Step2: Find the maximum value of the function

Substitute \(x = - 1\) into the function \(f(x)=-x^{2}-2x + 3\)
\(f(-1)=-(-1)^{2}-2\times(-1)+3\)
\(=-1 + 2+3\)
\(=4\)

Answer:

Price change: \(-1\) (This means a price decrease of \(\$1\)), Maximum profit per toy: \(4\)