*f* has a vertical asymptote at *x* = 0, and no holes. We can start drawing:
Since the degree of the numerator and denominator are the same, *f* has a horizontal asymptote at *y* = 3: *f* is 0 when its numerator is 0 and its denominator is not. This occurs when .
The only thing left is to find where *f* is negative and positive. When the numerator and denominator are both negative, therefore *f* is positive. When , the numerator is positive and the denominator is negative, so *f* is negative. We now have enough information to get a rough sketch of the piece of *f* that lies to the left of the vertical asymptote at *y* = 0. When *x* > 0, the numerator and denominator are both positive and *f* is positive. Since *f* must approach its asymptotes, *f* looks like this: We didn't need to worry about *f*(0) for this function, since *f* is undefined at 0. A number line is a useful tool for figuring out where a function is negative and positive, and we'll use this tool in one of the examples. Wherever *f* is 0 or undefined, its sign has the ability to change. Draw a number line and mark all the values of *x* where *f* is 0 or undefined. From this, find the sign of *f* in between those marked values. |