2. {15 p15) Analyze the function (SHOW ALL STEPS!) and sketch the graph following the steps from this video {or the book}: Graphing Function: steps and example: MW _ 2 y_x236 Graphing Guidelines for y = f(x) 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. 2. 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. 3. Exploit symmetry. Take advantage of symmetry. For example, is the function even (f(-x) = f(x)), odd (f(-x) = -f(x)) or neither? Periodicty: f(x+p)=f(x) 4. Locate all asymptotes and determine end behavior. Vertical asymptotes often occur at zeros of denominators. Hori- zontal asymptotes require examining limits as x -> + 0; 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. 5. Find the first and second derivatives. They are needed to determine extreme values, concavity, inflection points, and the intervals on which f is increasing or decreasing. Computing derivatives-particularly second derivatives-may not be practical, so some functions may need to be graphed without complete derivative information. 6. Find critical points and possible inflection points. Determine points at which f' (x) = 0 or f' is undefined. Determine points at which f"(x) = 0 or f" is undefined. 7. Find intervals on which the function is increasing/ decreasing and concave up / down. The first derivative deter- mines the intervals on which f is increasing or decreasing. The second derivative determines the intervals on which the function is concave up or concave down. 8. 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. 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 show all the essential details of the function