UChicago MATH 15200 - Graphing (16 pages)

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Graphing

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Graphing

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16
School:
University of Chicago
Course:
Math 15200 - Calculus-2
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GRAPHING MATH 152 SECTION 55 VIPUL NAIK Corresponding material in the book Section 4 8 Difficulty level Hard What students should definitely get The main concerns in graphing a function how to figure out what needs figuring out It is important for students to go through all the graphing examples in the book and do more hands on practice Transformations of graphs Quickly graphing constant linear quadratic graphs What students should hopefully get How all the issues of symmetry concavity inflections periodicity and derivative signs fit together in the grand scheme of graphing The qualitative characteristics of polynomial function and rational function graphs as well as graphs involving a mix of trigonometric and polynomial functions Weird feature Ironically there are very few pictures in this document The naive explanation is that I didn t have time to add many pictures The more sophisticated explanation is that since the purpose here is to review how to graph functions having actual pictures drawn perfectly is counterproductive Please keep a paper and pencil handy and sketch pictures as you feel the need Executive summary 0 1 Symmetry yet again Words 1 All mathematics is the study of symmetry well not all 2 One interesting kind of symmetry that we often see in the graph of a function is mirror symmetry about a vertical line This means that the graph of the function equals its reflection about the vertical line If the vertical line is x c and the function is f this is equivalent to asserting that f x f 2c x for all x in the domain or equivalently f c h f c h whenever c h is in the domain In particular the domain itself must be symmetric about c 3 A special case of mirror symmetry is the case of an even function An even function is a function with mirror symmetry about the y axis In other words f x f x for all x in the domain Even also implies that the domain should be symmetric about 0 4 Another interesting kind of symmetry that we often see in the graph of a function

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