question on plotting graph

Question

if we have the set of all ordered pairs $(x,x^2/y)$ such that x is inn $R$ an y is in $N$ then am I right in saying this is like the set of ordered pairs defined by $y=x^2$ and also $y=(1/2)x^2$ and so on so to plot this set on the x-y plane it would look like y=x^2 and then lots of the same stretched by scale factor 1/2 in the y direction?

Thanks

Answer 1

You are entirely correct, the set of the ordered pairs you describe is the union of the graphs of the functions $f_y: \mathbb{R} \to \mathbb{R}_+, x \mapsto \frac{x^2}{y}$ for $y \in \mathbb{N}$.

Here's a quick plot to support your intuition.

You should perhaps avoid saying the pairs are defined by $y = x^2$ though, as your parameter is already called $y$ and it might lead to confusion. Using my notation from above and describing the pairs in the form $(x,f_y(x))$ will be more practical if you need to work with them further.

Answer 2

Yes, the graph of the ordered pairs $(x,x^2/n)$ is the graph of $y=x^2/n$. It's graph would be like the graph of $y=x^2$ but shrunk by a factor of $1/n$ vertically. If $n$ is negative, it will also be reflected in the $x$-axis.