The background for graphs are one horizontal and one vertical axis. They meet in the middle of the graph at the origin. Along the right side of the horizontal axis, starting at the origin, the axis is marked with increasing positive numbers. Along the left side of the horizontal axis, starting at the origin and going to the left, the axis is marked with increasing negative numbers. On the top half of the vertical axis, starting at the origin and going up, the axis is marked with increasing positive numbers. On the bottom half of the vertical axis, starting at the origin and going down, the axis is marked with increasing negative numbers
For graphing equations, the vertical axis is the Y axis and the horizontal axis is the X axis. Functions of X are written Y = f(X). To graph a function, we need to find points (Xn, Yn) such that Yn equals the value of the function at the point when X = Xn. If the function is linear, you will only need to find two points. If the function is not linear, more points are needed. For non-linear functions, critical points like the places that the function crosses the axes and extrema -- maximums and minimums -- should be included.
For graphing complex numbers, let the vertical axis be imaginary and the horizontal axis be real. The point (a, b) represents the complex number a + bi. The interesting thing about graphing complex numbers is that a figure that is made of complex points can be easily moved or rotated by a simple multiplication. This can be done without using complex numbers, but it is much more computationally intensive. There is a 3D version of complex numbers called quaternions that can do the same thing in 3D. This is how computer generated graphics are manipulated.
It is possible to draw graphs in three dimensions with the addition of a third axis -- the Z axis -- that goes through the origin perpendicular to the X and Y axes. The Z axis comes straight out of the page with positive gradation above the page and negative gradations behind the page. Of course, the 3D graphs have to be drawn on the flat page, and it takes considerably more skill that drawing a flat graph.