# PLEASE USE MCV4U1 NOTATION AND NOT UNIVERSITY NOTATIONS

1. Use the steps below to guide you through your creation, analysis, and careful sketching by hand of a polynomial function. Use the algorithm for curve sketching attached to this question to guide you as well.

a) Follow all steps for the cubic equation: f(x) = 2x^3 + x^2 - 13x + 6 = 0

b) Use the Algorithm for Curve Sketching as a guide for your entire process. Remember that not all steps will be applicable for a polynomial function.

c) Include a table to help in your analysis for steps 7 and 8 (first derivative work), and then a therefore statement that communicates what you learned.

d) Include a table to help in your analysis for steps 9 and 10 (second derivative work), and then a therefore statement that communicates what you learned.

e) The culmination of all your work will be an accurate, hand-drawn, fully-labelled, sketch of your function

Image transcriptions

MCV4U: Algorithm for Curve Sketching and Some Practice Waterloo's General Algorithm for Curve Sketching Ms Spring's Step i: Take some time to just look at f(x). Maybe you already know some things about this type of function. Step ii: It might be helpful to factor the parts of f(x) if possible. This is a great guide, 1. Find the domain of f(z); i.e., the set of a-values for which f(z) is defined. but we don't always need 2. Find the y-intercept: (0, f(0)) You are not locked into this particular order. For every single step. 3. Determine any discontinuities of f. example, it might be easier to do step 1 after step 3. Why? 4. Find the z-intercept(s) (z, 0): f(z) = 0 For example, steps 3,5,6 will 5. Explore lim f(x) and lim f(z) to determine whether horizontal asymptotes exist, and if not be necessary for a polynomial 2-+400 function. Why not? so, whether y - f(x) approaches the asymptote from above or below. 6. If f(z) is a rational function where the degree of the numerator is exactly 1 greater than the We could already do many degree of the denominator, then find the oblique asymptote and determine how it is approached of these steps in Advanced as x -+ +oo and as a + -co. Functions, but steps 7-10 7. Find any critical points where f'(e) - 0 or f'(c) is undefined for c in the domain of f are new "Calculus" tools that help so much in sketching an 8. Determine the intervals where f(x) is increasing and where f(x) is decreasing by checking accurate graph. whether f'(z) changes sign at any critical point of f(z), or at any discontinuity of f (z). It f'(z) changes sign at a critical point a = c, then a local extreme exists at (c, f(c)) provided that Tip: Using tables for steps 7-10 f(z) is continuous at z = c. is really helpful! 9. Determine all points where f"(x) - 0 or f"(x) does not exist in order to locate the points at which f(z) may change concavity. Note that this will include any points where f (z) or f'(z) is undefined or where f(a) is discontinuous 10. By checking the sign of f"(z) on the intervals determined by the points found in step 9), determine the intervals where f(z) is concave upward and the intervals where f(z) is concave downward. Locate any points of inflection. 11. Sketch the graph of y = f(z), incorporating all the above information.

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