Copyright © University of Cambridge. All rights reserved.

## 'Curve Fitter 2' printed from http://nrich.maths.org/

In Cartesian coordinates, the squared distance $D^2$ beween two
points $(a, b)$ and $(c, d)$ is given by

$$

D^2 = (a-c)^2+(b-d)^2

$$

Imagine that the function $y=f(x)$ has a squared distance $D^2$
between two of its turning points. Which of the following functions
also definitely have a
squared distance $D^2$ between two of their turning points? ($a$ is
a constant)

$$y=f(x)-a\quad\quad y=af(x)\quad\quad y = f(ax)\quad\quad
y=f(x-a)$$

Geometrically, what do these transformed equations correspond
to?

Use these ideas to help you to find a cubic equation for which the
squared distance $D^2$ between the turning points is a whole
number.

Very hard extension: Can you find
a cubic equation for which the distance between the turning points
is a whole number (Warning: we can't find a simple solution to
this, so be prepared for an exploration!)