Question

What are the asymptotes of `y=1/x-2` and how do you graph the function?

Answer 1

The most useful thing when trying to draw graphs is to test the zeroes of the function to get some points that can guide your sketch.

Consider `x = 0`:
`y = 1/x - 2`
Since `x=0` cannot be substituted in directly (since it's in the denominator), we can consider the limit of the function as `x->0`. As `x->0`, `y->\infty`. This tells us that the graph blows up to infinity as we approach the y-axis. Since it will never touch the y-axis, the y-axis is a vertical asymptote.

Consider `y = 0`:
`0 = 1/x - 2`
`x= 1/2`

So we have identified a point that the graph passes through: `(1/2,0)`

Another extreme point we can consider is `x->\infty`. If `x->+\infty`, `y-> -2`. If `x->-\infty`, `y->-2`. So at both ends of the x-axis, y will approach -2. This means there is a horizontal asymptote at `y=-2`.

So we have found out the following:
Vertical asymptote at `x=0`.
Horizontal asymptote at `y=-2`.
Point contained in graph: `(1/2,0)`.

graph{1/x -2 [-10, 10, -5, 5]} You should notice that all three of these facts provide enough information to draw the graph above.