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

Mar 1, 2018

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 $x = 0$:
$y = \frac{1}{x} - 2$
Since $x = 0$ cannot be substituted in directly (since it's in the denominator), we can consider the limit of the function as $x \to 0$. As $x \to 0$, $y \to \setminus \infty$. 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 $y = 0$:
$0 = \frac{1}{x} - 2$
$x = \frac{1}{2}$

So we have identified a point that the graph passes through: $\left(\frac{1}{2} , 0\right)$

Another extreme point we can consider is $x \to \setminus \infty$. If $x \to + \setminus \infty$, $y \to - 2$. If $x \to - \setminus \infty$, $y \to - 2$. 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: $\left(\frac{1}{2} , 0\right)$.

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.