Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.
In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.
Sketch the graphs for this implicitly defined family of functions.
Two functions $f(x)$ and $g(x)$ were plotted on the same axes, where $$ f(x) =\left(\frac{a}{x}\right)^x\quad \quad g(x) = b\exp\left(-\frac{(x-c)^2}{d}\right) $$ I chose the coefficients $a, b, c$ and $d$ so as to make the function $g(x)$ match $f(x)$ 'as closely as possible' for points past the maximum of $f(x)$. My resulting charts were as follows. Is it possible to approximately work out the values I chose? Can you choose values to obtain a closer match between the two?