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**Graphing by hand**

- Determine the domain of the function. Distinguish between holes which are factors that can be canceled and vertical asymptotes, which are factors that cannot. Draw the vertical asymptotes as dashed lines.
- Identify the end behavior of the function by comparing the degrees of the numerator and denominator and determine if there exists a horizontal or oblique asymptote. Draw the horizontal or oblique asymptotes as dashed lines.
- Identify the holes of the function and note them with empty circles.
- Identify the zeroes and intercepts of the function and plot them.
- Use the sign test to determine the behavior of the function near the vertical asymptotes and holes.
- Connect everything as best you can.

**Things to remember**

- Horizontal asymptotes can be crossed
- Zeroes means the function touches the x-axis, not necessarily crosses it
- Clearly distinguish whether a point is a hole, zero, or vertical asymptote
- Test all possible number situations when sign testing.

Practice problems can be found here.