Graphing a function means to visibly mark within an N-dimensional space all points that fulfill a certain quality.

The general idea within 2-dimensions is to first have concept of rectangular or Cartesian coordinates. In such a system, the input to the function denote the distance x along a straight line (called xa upon which we will form a motion) and the output denote the y distance of the motion along another line perpendicular to the first (called ya). Then, the point we consider within the graph of the function is the intersections of the two lines xay and yax, where xay is the line resulting from translating xa by y in the direction of ya. Similarly, yax is given by translating ya by x in the direction of xa.

Yes, the above sounds incredibly weird and esoteric; however, it is the more rigorous way of graphing a point. More simply put, one graphs a Cartesian point in a plane by drawing an infinitely dense grid of lines in the plane and then letting the outputs to the function denote the height of the curve at any point whilst moving down one particular line in the grid. The grid allows amateur students to draw the curves properly.

In order to graph in coordinate systems that are not Cartesian, one has to first get the Cartesian graph and then apply a transformation to every point in the curve. The result of a transformation R^2 -> R^2 is a new curve in the plane representing the graph of the function in some other coordinate system.

Another way to graph in alternate coordinate systems is to use the rigorous method described above in non-Euclidean geometries. The only danger with this is that in (for instance) Spherical Geometry, lines intersect in two locations. Picking strictly one intersecting point yields the system known as unit Spherical Coordinates.

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