What Is The Optimal Price?

An example of a quadratic function that can be used to answer questions is in the example of a question about the optimal price for a product to be. This sort of question could be written something like this:

"Nigel sells 200 really delicious cookies for 2 dollars each. For every 20 cents he increases the price, he sells 10 fewer cookies. At what price would Nigel make the most money selling cookies?"

The amount of money that Nigel makes from selling 200 cookies is a simple multiplication formula:

$$ 200\times2=400 $$

How much a change in price changes how many cookies are sold however can be expressed as a quadratic. Since the loss of the sales of 10 cookies scales with the increase in price of 20 cents, we can define the price increase of 20 cents as $x$. This gives values of $x$ and their coefficients we can add to our equation.

$$ y=(-10x+200)(0.2x+2) $$

$x$ is the increase in the price by 20 cents, each one of which reduces the number of sales by 10 (as seen in the left hand bracket, with the $-10x$), but increases the price per cookie sold by 20 cents (as in the right hand bracket, with the $0.2x$). This turns the equation into a quadratic in the factored form, which can be expanded to create a quadratic function:

$$ y=-2x^2+20x+400 $$

With our equation in the standard form, we can convert it to the vertex form, in order to find the vertex, which in this function is the point where the maximum revenue can be earned, given $x$ number of increases in the price by 20 cents. Following the method to convert standard into vertex form we can get the vertex form of:

$$ y=-2(x-5)^2+450 $$

From this vertex form, we can see that the $x$ value that gives the maximum point on the quadratic function is +5 (the inverse of $h$) while the $y$ coordinate of the vertex is 450. This means that by increasing the price to 3 dollars (an increase of $5x$ where $x$ is an increase of 20 cents), Nigel can make 450 dollars selling cookies.