Simple function transformation question and cusp question

Hey, I'm just trying to memorize function transformations so I can picture graphs in my head, and I read that multiplying a function (or y) by a number will cause it to stretch vertically, however I found that this isn't true for something like f(x) = x^2. If you do 2y = x^2, it seems to stretch the graph horizontally. This coincides with the rule that says if you multiply the input of a graph between 0 and 1, it will stretch it. Also, 2y = x^2 is equal to y=(x^2)/2 and y=(0.5)(x^2). So what should I try to remember here? It seems that the rule where it's supposed to stretch vertically isn't applying here.

Second question, I was reading that cusps cannot have a derivative. It makes sense that discontinuities and vertical lines have no derivatives because a slope does not exist, but if I look at a picture of a cusp, it looks like a horizontal line could sit on the peak tangent to it. Am I picturing this wrong?

For the question on cusps. I like to think of cusps as two curves meeting at the same end point. Like say you have a line that curves down and stops at say x=3. that point at x=3 does not have a derivative, because it is an end-point right? Cusps are essentially the same principle, but with two curves coming to the same end point.

Its easy to think of there being a horizontal tangent, but there could be a tangent at any slope at the point, rotating it around, which i believe is why it is undefined.