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Q Graphing Guidelines for y = f(x) Accoun 1. Use $4.4 guideline to sketch the curve: y = x - 4x1/3. 1. Identify the domain or interval of interest. On what interval(s) should the function be graphed? It may be the domain of the function or some subset of the domain. A). Domain, Symmetry, Asymptotes, Intercepts, 2. Exploit symmetry. Take advantage of symmetry. For example, is the function even (f(-x) = f(x)), odd (f(-x) = -f(x)), Dashboa or neither? B). f', Critical Points and Local Max/Min Values, 3. Find the first and second derivatives. They are needed to determine extreme values, concavity, inflection points, and intervals of increase and decrease. Computing derivatives-particularly second derivatives-may not be practical, so some functions may need to be graphed without complete derivative information. Course C). f", Concavity and Points of Inflection , 4. Find critical points and possible inflection points. Determine points at which f'(x) = 0 or f' is undefined. D). List VIP Points, and Graphing. Determine points at which f"(x) = 0 or f" is undefined. 5. Find intervals on which the function is increasing / decreasing and concave up / down. The first derivative deter- Calenda mines the intervals of increase and decrease. The second derivative determines the intervals on which the function is concave up or concave down. 6. Identify extreme values and inflection points. Use either the First or Second Derivative Test to classify the critical points. Both x- and y-coordinates of maxima, minima, and inflection points are needed for graphing. 7. Locate all asymptotes and determine end behavior. Vertical asymptotes often occur at zeros of denominators. Hori- zontal asymptotes require examining limits as x -> + co; these limits determine end behavior. Slant asymptotes occur with rational functions in which the degree of the numerator is one more than the degree of the denominator. Histor 8. Find the intercepts. The y-intercept of the graph is found by setting x = 0. The x-intercepts are found by solving f(x) = 0; they are the real zeros (or roots) of f. ? 9. Choose an appropriate graphing window and plot a graph. Use the results of the previous steps to graph the function. If you use graphing software, check for consistency with your analytical work. Is your graph complete-that is, does it Help show all the essential details of the function