Challenge Level

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.