Final answer:
To sketch a rational function, identify x/y-intercepts, determine vertical/horizontal/slant asymptotes, plot additional points for graph shape, and sketch the graph with scaled axes and correct asymptote behavior.
Step-by-step explanation:
To sketch the graph of a rational function by hand, you can follow these steps:
- Identify and plot any x-intercepts by setting the numerator equal to zero and solving for x.
- Find the y-intercept by evaluating the function at x = 0, if possible.
- Determine the vertical asymptotes by setting the denominator equal to zero and solving for x.
- Locate any horizontal asymptotes by comparing the degrees of the numerator and denominator or by finding the limit of the function as x approaches positive or negative infinity.
- Analyze the function to find any slant asymptotes, which occur if the degree of the numerator is exactly one greater than the degree of the denominator.
- Plot additional points to get a sense of the shape of the graph, especially near the asymptotes.
- Sketch the graph using a ruler and pencil, making sure to scale the axes and to approach asymptotes correctly.