Find the asymptotes and represent them with a dashed line. Choose graphing points to include several near the asymptote and graph in the normal way. For example, when graphing the function y = 1/(x -3), there is an asymptote at x = 3, because y is undefined when x = 3. Choosing points near the asymptote we find 1/(3.01 - 3) = 100 and 1/(2.99 - 3) = -100. The graph goes up along the right side of the asymptote and down along the left side of the asymptote.
Move the graph around the page. Modifying the x in the equation moves the graph left and right. So adding 2 to the x in an equation moves it two places. Y = 1/((X + 2) - 3 = 1/(X -1). The graph of y = 1/(X -1) looks exactly like the graph of Y = 1/(X - 3) except it is shifted two units to the left. Similarly, adding 2 to the y part gives (y + 2) = 1/(x - 3) or y = 1/(x - 3) - 2 = (7 - 2x)/3 - 5 which is the same graph shifted two units down. Subtracting a distance from the Y shifts the entire graph up on the page.
Scale the graph to fit the page. The general formula for vertical stretching and shrinking a formula y = f(x) is y = af(x) when a > 0. Stretching takes place where a > 1; shrinking takes place when a < 1. So the graph of y = 1/(x-3) is doubled in the vertical direction if you graph y = 2/(X-3) instead, and shrunk by half in the vertical direction if you graph y = 0.5/(x - 3). Horizontal scaling of y = f(x) is accomplished by y = f(bx). To stretch y = 1/(x-3) horizontally use y = 1/(2X - 3); to shrink it use y = 1/(0.5x - 3).