Can you massage the parameters of these curves to make them match as closely as possible?

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?