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.
Investigate the family of graphs given by the equation x^3+y^3=3axy
for different values of the constant a.
Sketch graphs of
$$y = \left[1 + (x - t)^2\right]\left[1 + (x + t)^2\right]$$
for $t = -1/2$, $1/2$ and $2$. You will see that these graphs have 'different shapes'. Suppose the parameter $t$ varies, then the general shape of the graph varies continuously with $t$. Show that the graph always has a shape similar to the examples above and find the values of $t$ at which there are transitions from one shape to another.