Why do polynomial equations make the shapes that they do? I will use two examples.
$$
y=(x-1)(x-2)(x-3)
$$
Why does it go up then down then up again?
$$
y=(x-1)(x-2)(x-3)(x-4)(x-5)
$$
Why does that one have two large peaks/troughs and two small ones?
Thank you for any help you are willing to give.

Its actually depends on the degree of the polynomial. So if you have $x^4$ in your equation then you would have a U shaped curve. There are many patterns you can explore by playing with coefficients and powers of $x$
