People think of "drawing a graph of a function" as drawing a line, so if y=sin(x) they end up with a wriggly line going infinitely far left and right.

What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being:

  { (x,y) : y=sin(x) is true }

Realising that opens the doors to equivalences between different ways of thinking. We can think of a permutation of objects as both the act of permuting them, and as the result of applying that permutation to the default initial position. We can think of a vector (4,6,9) as a location in space, and as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a location on the number line, or as the action of adding 3 to something, or as the action of multiplying 3 my something, and so on.

We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power.

Does that help?