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The amount of profit Bill makes per toy when he increases or decreases the price of his handmade toys can be modeled by the function f(x) = –x2 – 2x + 3. What price change gives him the highest profit? What is the highest profit per toy?

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Answer:

The profit equation is modeled by the equation:

f(x) = -x^2 - 2*x + 3

The first thing that you can notice, is that the leading coefficient is negative, then the arms of the quadratic function will go downwards. This implies that the maximum of this function will be in the vertex.

For a quadratic equation

a*x^2 + b*x + c

the vertex is at:

x = -b/2a

In our case, the vertex is at:

x = +2/(-2) = -1

we evaluate our function in this value:

f(-1) = -(-1)^2 - 2*(-1) + 3 = -1 + 2 + 3 = 4

So x = -1 means that he must decrease the price by one unit, and the highest profit by toy will be 4.

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