3 Parabolas and optimisation

Because the vertex may represent a highest or lowest point, a quadratic function may be the appropriate type of function to choose in a modelling problem where a maximum or a minimum is involved (optimisation problems for example). Consider the problem of working out the selling price for the product of a cottage industry that would maximise the profit, given certain details of costs and assumptions about market behaviour. A possible function relating profit $\left(\pounds M\right)$ to selling price $\left(\pounds P\right)$ , is

$M=-10P^2+320P-2420\left(12\le P\le 20\right).$

Note that this is a quadratic function. By comparing this function with the form $y=a{x}^2+bx+c$ it is possible to decide whether the corresponding parabola that would result from graphing $M$ against $P$ , would open upwards or downwards. Here $M$ corresponds to $y$ and $P$ to $x$ . The coefficient corresponding to $a$ in the general form is $-10$ . This is negative, so the resulting parabola will open downwards. In other words it will have a highest point or maximum for some value of $P$ . This is comforting in the context of an optimisation problem! We can go further in specifying the resulting parabola by reference to the other general form: $y-C=A{\left(x-B\right)}^2.$ If we multiply out the bracket on the right hand side we get (as seen at the beginning of HELM booklet  5.2)

$y-C=A{x}^2-2ABx+A{B}^2$

or

$y=A{x}^2-2ABx+A{B}^2+C.$

Comparing this general form with the function relating profit and price for the cottage industry:

$y=A{x}^2-2ABx+A{B}^2+C$

$\downarrow \downarrow \downarrow$

$M=-10P^2+320P-2420$

Using the equivalances suggested by the arrows, we see that

$A=-10,$

$2AB=-320$

$A{B}^2+C=-2420.$

These are three equations for three unknowns. Putting $A=-10$ in the second equation gives $B=16.$ Putting $A=-10$ and $B=16$ in the third equation gives

$-2560+C=-2420,$

and so

$C=140.$

This means that the equation for $M$ may also be written in the form

$M-140=-10{\left(P-16\right)}^2,$

corresponding to the general form $y-C=A{\left(x-B\right)}^2$ . In the general form, $C$ corresponds to the value of $y$ at the vertex of the parabola. Since $y$ in the general form corresponds to $M$ in the current modelling context, we deduce that $M=140$ at the highest point on the parabola. $B$ represents the value of $x$ at the lowest or highest point of the general parabola. Here $x$ corresponds to $P$ , so we deduce thet $P=16$ at the vertex of the parabola corresponding to the function relating profit and price. These deductions mean that a maximum profit of £ $140$ is obtained when the selling price is £ $16$ .