Quadratic models in real-world contexts

A quadratic function can model a quantity that rises and then falls, such as the height of a jet of water, the profit at a given price, or the shape of an arch. The roots give where (or when) the quantity is \(0\), and the turning point, halfway between the roots, gives its greatest or least value.

Method

1. To find the value for a given input, substitute it into the model.
2. To find when the quantity has a given value, set the model equal to that value, rearrange to \(0\) and factorise.
3. To find where (or when) the quantity is \(0\), set the model equal to \(0\).
4. For the greatest (or least) value, take the input halfway between the roots and substitute it. Interpret every answer in the context and reject impossible values.

Example

A tunnel entrance has height \(h=4x-\frac{1}{2}x^2\) metres at \(x\) metres from one side. Find its width and greatest height, and the width of the part of the tunnel that is at least \(6\) m high.

\(\begin{aligned} 4x-\frac{1}{2}x^2 &= 6 \\ x^2-8x+12 &= 0 \\ (x-2)(x-6) &= 0 \\ x=2 \quad &\text{or}\quad x=6 \end{aligned}\)

Width: \(\frac{1}{2}x(8-x)=0\) gives \(x=0\) or \(x=8\), so the tunnel is \(8\) m wide. Greatest height at \(x=4\): \(h=16-8=8\) m. The tunnel is at least \(6\) m high from \(x=2\) to \(x=6\), a width of \(4\) m.

Check: \(h(2)=8-2=6\) and \(h(6)=24-18=6\). A common error is to give only one of the two values; a height below the top is reached on both sides of the line of symmetry.