When you are sketching the graph of a function, either by hand or with a graphing utility, remember that normally you cannot show the entire graph. The decision as to which part of the graph you choose to show is often crucial.
Guidelines for analyzing the graph of a function
1. Determine the domain and range of the function
2. Determine the intercepts, asymptotes, and symmetry of the graph
3. Locate the x-values for which f’x and f’’x either are zero or do not exist. Use the results to determine relative extrema and points of inflection
To do curve sketching you must use the following skills and techniques
1. X-intercepts and y-intercepts
2. Symmetry
3. Domain and range
4. Continuity
5. Vertical asymptotes
6. Differentiability
7. Relative extrema
8. Concavity
9. Points of inflection
10. Horizontal asymptotes
11. Infinite limits at infinity
When curve sketching you must first take the derivative, then you must take the second derivative. Then you must find the x-intercepts, and then find the y-intercept. Then find the vertical asymptotes. Then find the horizontal asymptotes. Then you use your derivative to find the critical numbers. Then use your critical numbers to find your points of inflection. Then you must find your domain and symmetry. And then you must test your intervals.
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