Executive summary0.1. Symmetry yet again0.2. Graphing a function1. Graphing in general1.1. Graphs -- utility, sketching and plotting1.2. The domain of a functionIntercepts and a bit of plotting1.3. Symmetry/periodicity1.4. First derivative1.5. Second derivative1.6. Classifying and understanding points of interest1.7. Sketching the graph2. Graphing particular functions2.1. Constant and linear functions2.2. Quadratic functions2.3. Cubic functions2.4. Polynomials of higher degree2.5. Rational functions: the many concerns2.6. Piecewise functions2.7. A max-of-two-functions example2.8. Trigonometric functions2.9. Mix of polynomial and trigonometric functions2.10. Functions involving square roots and fractional powersA more complicated version of the coffee shop problem3. Subtle issues3.1. Equation-solving troubles3.2. Graphing multiple functions together3.3. Transformations of functions/graphs3.4. Can a graph be used to prove things about a function?3.5. Sketching curves that are not graphs of functions3.6. Piecewise descriptions, absolute values and max/min of two functions4. Addenda4.1. Addendum: Plotting graphs using MathematicaAddendum: using a graphing software or graphing calculatorGRAPHINGMATH 152, SECTION 55 (VIPUL NAIK)Corresponding material in the book: Section 4.8Difficulty level: Hard.What students should definitely get: The main concerns in graphing a function, how to figure outwhat needs figuring out. It is important for students to go through all the graphing examples in the bookand do more hands-on practice. Transformations of graphs. Quickly graphing constant, linear, quadraticgraphs.What students should hopefully get: How all the issues of symmetry, concavity, inflections, period-icity, and derivative signs fit together in the grand scheme of graphing. The qualitative characteristics ofpolynomial function and rational function graphs, as well as graphs involving a mix of trigonometric andpolynomial functions.Weird feature: Ironically, there are very few pictures in this document. The naive explanation is that Ididn’t have time to add many pictures. The more sophisticated explanation is that since the purpose here isto review how to graph functions, having actual pictures drawn perfectly is counterproductive. Please keepa paper and pencil handy and sketch pictures as you feel the need.Executive summary0.1. 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 symmetryabout a vertical line. This means that the graph of the function equals its reflection about thevertical line. If the vertical line is x = c and the function is f, this is equivalent to asserting thatf(x) = f(2c − x) for all x in the domain, or equivalently, f(c + h) = f(c − h) whenever c + h is inthe 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 functionwith 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 is half-turnsymmetry about a point on the graph. This means that the graph equals the figure obtained byrotating it by an angle of π about that point. A point (c, d) is a point of half-turn symmetry iff(x) + f(2c − x) = 2d for all x in the domain. In particular, the domain itself must be symmetricabout c. If f is defined at c, then d = f(c).(5) A special case of half-turn symmetry is an odd function, which is a function having half-turn sym-metry about the origin.(6) Another symmetry is translation symmetry. A function is periodic if there exists h > 0 such thatf(x + h) = f (x) for all x in the domain of the function (in particular, the domain itself should beinvariant under translation by h). If a smallest such h exists, then such an h is termed the period off.(7) A related notion is that of a function with periodic derivative. If f is differentiable for all realnumbers, and f0is periodic with period h, then f(x + h) − f (x) is constant. If this constant valueis k, then the graph of f has a two-dimensional translational symmetry by (h, k) and its multiples.Cute facts...(1) Constant functions enjoy mirror symmetry about every vertical line and half-turn symmetry aboutevery point on the graph (can’t get better).1(2) Nonconstant linear functions enjoy half-turn symmetry about every point on their graph. They donot enjoy any mirror symmetry because they are everywhere increasing or everywhere decreasing.(3) Quadratic (nonlinear) functions enjoy mirror symmetry about the line passing through the vertex(which is the unique absolute maximum/minimum, depending on the sign of the leading coefficient).They do not enjoy any half-turn symmetry.(4) Cubic functions enjoy half-turn symmetry about the point of inflection, and no mirror symmetry.Either the first derivative does not change sign anywhere, or it becomes zero at exactly one point, orthere is exactly one local maximum and one local minimum, symmetric about the point of inflection.(5) Functions of higher degree do not necessarily have either half-turn symmetry or mirror symmetry.(6) More generally, we can say the following for sure: a nonconstant polynomial of even degree greaterthan zero can have at most one line of mirror symmetry and no point of half-turn symmetry. Anonconstant polynomial of odd degree greater than one can have at most one point of half-turnsymmetry and no line of mirror symmetry.(7) If a function is continuously differentiable and the first derivative has only finitely many zeros inany bounded interval, then the intersection of its graph with any vertical line of mirror symmetryis a point of local maximum or local minimum. The converse does not hold, i.e., points where localextreme values are attained do not usually give axes of mirror symmetry.(8) If a function is twice differentiable and the second derivative has only finitely many zeros in anybounded interval, then any point of half-turn symmetry is a point of inflection. The converse doesnot hold, i.e., points of inflection do not usually give rise to half-turn symmetries.(9) The sine function is an example of a function where the points of inflection and the points of half-turnsymmetry are the same: the multiples of π. Similarly, the points