For this question you may like to use a computer graph drawing
application or a graphic calculator. For $t = -1/2$, $1/2$ and $2$,
sketch graphs of $y = [1 + (x - t)^2][1 + (x + t)^2]$.
You can download the shareware program Graphmatica for free from
here
as NRICH is an approved distributor of this program. You can find
more information about the program from
http://www.graphmatica.com/
Then try other values of the parameter $t$. You will see that these
graphs have 'different shapes'. Suppose the parameter $t$ varies,
then the general shape of the graph varies continuously with
$t$.
You can get this far without calculus but you'll probably need
calculus to find the turning points and to show that the graph
always has a shape similar to the examples above and to find the
values of $t$ at which there are transitions from one shape to
another.