Curve Sketching

In this topic:

    1. State the ket steps in sketching a graph.
    
    2. Illustrate with the functions and
    

The following procedure can be used to graph many commonly used function.

    1. If given a function, determine the domain and range.
    
    2. If possible algebraically simplify the function.

    3. Determine any symmetry. Test for symmetry with respect to the -axis, -axis, and the origin.
    
    4. Determine any -intercepts and -intercepts.

    5. Determine any vertical asymptotes.

    6. Determine any horizontal asymptotes.

    7. Determine the first order critical numbers.

    8. Apply the First Derivative Test. This test will determine intervals of increasing and decreasing and any relative extremum.

    9. Apply the Second Derivative Test. This test will determine any relative extremum, but may possible fail.

    10. Determine the second order critical numbers.

    11. Apply the Concavity Test. This will determine any points of inflection and intervals of concavity.

    12. Determine any vertical tangents.

    13. Determine any cusps.
    
    14. Plot Points.
    
    15. Sketch the curve.

Example (Curve Sketching) Sketch the graph of the rational function



showing all special features.

    Solution. The domain is



The and intercepts are both Since the function is even. The curve is symmetric about the -axis.



Therefore, the line is a horizontal asymptote. Since the denominator is 0 when we compute the following limits:



Therefore, the lines and are vertical asymptotes.

Next we find the derivative function.



Since when and when is increasing on and and decreasing on and The only critical number is Since changes sign from positive to negative at is a local maximum by the First Derivative Test. Also,



Since for all we have



and



Thus the curve is concave downward on the intervals and and concave downward on There is no point of inflection since and are not in the domain of Here is a sketch of the graph:


  

Example (Curve Sketching) Sketch the graph of the trigonometric function



showing all special features.

    Solution. The domain is The -intercept is The -intercept occur when



which is precisely when and , because we need only consider since function is periodic via,  



There are no asymptotes. Computing the derivative,







Thus, when or when so in when consider the critical number and and Applying the First Derivative Test we find:



Computing



we find the second order critical numbers as and where and Applying the Concavity Test we find,



Here is a sketch of the graph:

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Cite this as:
Curve Sketching
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/curve-sketching.html