Reciprocal Graphs

y = 3/x:

The graph is split into two parts. Each part consists of 2 asymptotes, 1 towards the y-axis and 1 towards the x-axis. When y approaches +-∞, x approaches 0. When x approaches +-∞, y approaches 0. Each asymptote has a turning point, when the horizontal asymptote starts becoming a vertical asymptote and vice versa. There is a line of symmetry, which cuts through the 2 asymptotes.

y = 3/x + 4

This graph has about the same characteristics as the previous graph. Let us now overlay the two graphs to see the difference between them.

As it can be seen, the 2nd graph was "raised" by approximately 4 points. As such, we can see that any extra values added to the original equation has the effect of raising the graph.

Let us try another graph.

y = -1/x

This graph has approximately the same characteristics, though the parts of the graph lie in different regions.

y = -1/x - 4

The graph has the same characteristics, and it can be seen that it is "lowered" by 4. Now, we can also see that subtracting values also affects the graph.

Y = 0/X

y = 0/x is also y = 0, which is a straight line along the x-axis. At any point, y = 0.

y = 1/(x-2)

The graph is shifted slightly to the right, causing the bottom asymptote to intersect the y-axis. Otherwise, the graph has the same characteristics.

Let us over lay a line x = 2 on the graph.

As can be seen, the vertical asymptotes are towards the line x = 2. As y approaches +-∞, x approaches 2, but does not reach 2.