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Science, Technology, Engineering and Mathematics
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Curve Fitter 2
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
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!)
Maths Supporting SET
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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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