Final Video Summative In this task you will be using the curve sketching algorithm to sketch the graph of a function g(x) and you will demonstrate your knowledge by creating a video submission of the solution. Steps to Creating a Video Solution Pg. 1 Function Options: Pg. 2 Algorithm for Curve Sketching Pg. 3 Rubric: Pg. 4 Steps to Creating a Video Solution 1) Pick one (1) of the functions provided (Pg. 2) and apply the curve sketching algorithm to sketch the curve. You must include full algebraic solutions for each step. o Complete each step of the curve sketching algorithm o Include full algebraic solutions for each step Display the overall solution to the problem in an organized and clear manner Use any of the following to organize your solution: Headings, Numbering of steps and/or colour Ensure you make conclusions about each step on paper, indicating whether your findings seem reasonable. 2) Plan your video o The video should not be more than 5 minutes in length. 1 minute: Introduce the function and an overview of the procedure that you will undertake. * 3 minutes: Explain each of your algebraic steps as well as the results of each step. Do not simply read your solution out loud step by step. Instead, explain the steps in general before specifically stating the final result of each step. . Example: "To find the local maxima and minima where the tangent to the function has a slope of zero, I took the first derivative of the function using the product rule and set the derivative equal to 0 to solve for x. I checked each value to determine if it was a max or min by..etc... I therefore found that there is a local max at x = 0 and a local min at x = Avoid explaining algebraic details in too much detail Example: Rather than, "I then added 2 to both sides and divided both sides by 5", instead say, "I simplified the expression" or "I isolated the variable x". Use proper mathematical terminology from our course in your explanations. Your solutions should reference the terms: first derivative, second derivative, local maxima, local minima, end behaviour, inflection points, asymptote, intervals of increase/decrease, intervals of concavity etc. O 1 minute: Summarize your full solution. Show how your findings relate to the key points on your sketched graph and locations of zeros, maxima/minima, intervals of increase / decrease, asymptotes, intervals of concavity, etc. Page 1Pileggi - MCV 4UE Explain why your answer is appropriate and reasonable. Example, "It is reasonable that I only found two extreme values since a third degree polynomial has a maximum of 2 turning points." 3) Video Recording & Submission. o Keep your solution to under 5 minutes (do not record yourself solving the problem in real-time) O Speak clearly and with fluency; ensure all of your words are clearly audible. Video quality must be such that the solution is easily readable. You may have to do a couple of takes to get it right. When you're satisfied, submit it to the dropbox on brightspace. One way to produce a video solution: You can use free screen-recording software such as Screencastify Take a clear photo of your solutions (or produce them electronically) then upload them to Google Docs or Slides and record your screen while you scroll through and explain your steps. You may take a different approach - this is just a suggestion if you're not sure how to get started. Function Options: Pick one (1) function below and use the algorithm for curve sketching (above) to sketch the graph of g(x). g(x) = 2x - &x 1. 3x + 1 g(x) = X 12-4 2. 8(x) = 9+ x2 3 -12 3. g(x) = 2x-+ 4x (x+2) (x - 1)2Algorithm for Curve Sketching 1. Investigate the Original Function f(x) DOMAIN & INTERCEPTS - Find Domain, x and y-intercepts ASYMPTOTES - Find the domain Vertical Asymptotes Set denominator of / (x) = 0 BUT make sure the numerator is NOT zero (it may be a hole) Find the limit of f(x) as x-+ vertical asymptote values from both the sides Horizontal Asymptotes Find x- too /(x). There are 3 cases: 1. The degree of numerator is larger then the denominator 1+5 f(x) = *+ = No Horizontal Asymptote 2. The degree of the numerator is smaller then the denominator f(x) = => Horizontal Asymptote is y = 0 x+5 3. The degree of the numerator is the same as the denominator f(x) = + # H.A. is coefficients of the highest degrees -2x-+5 from numerator and denominator. y = Oblique Asymptotes Occurs when the numerator is one degree higher than the denominator. Use long division to find the oblique asymptote y = mx + b Sub a large positive and large negative number into both the oblique asymptote and into the original equation to see if the graph approaches from above or below. 2. FIRST DERIVATIVE TEST - Critical Points - Local Max/Min f(x) . where / (x) = 0 and/or / (x) DNE when f'(x) DNE we get a vertical tangent or a cusp analyze behaviour of / (x) on either side of critical points using a chart to find intervals of increase/decrease Critical Values f(x) + 0 0 + MAX f(x) ( c,. f (c,)) MIN (Cy f(c2)) 3. SECOND DERIVATIVE TEST - CONCAVITY f" (x) Page 3Pileggi - MCV 4UE Find where f (x) = 0 or "(x) DNE and check behaviour with a table Critical Values f" (x) + 0 ONE + f(x) LP. (C. f (C,)) Asymptote X = C Rubric: Criteria R 2 3 4 Verbal Analysis Introduction Introduction and Introduction and Focused Focused and and/or overall analysis plan is analysis is introduction and thorough introduction, analysis plan is incomplete and/or somewhat focused, analysis plan that introduction and description and not present is limited in clarity. some of the states the analysis plan that analysis of the objectives of objectives of the clearly states the problem solving the problem problem and the objectives of the are present but with approach that will problem and the misconceptions be taken. approach that will be taken Verbal Problem Has not yet Explanation shows Explains shows Explanation Thoroughly Solving selected the limited ability to some ability to sulliciently explains the most tools and/or select appropriate select appropriate demonstrated the appropriate tools selecting tools appropriate tools and apply the tools and apply the appropriate for applying the explaining algorithm algorithms to steps of the steps of the selection of tools algorithm. USE algorithm. algorithm. and the applied algorithm. Written Execution Has not yet Incorrect or Generally correct Content is fully Content is fully demonstrated incomplete content and complete correct and correct, complete content, appropriate with limited use of content, with some complete. Clear and relevant to the anthmeticlalgebraic written visuals and/or use of visuals or visuals and presentation. procedures, execution of a organization of organization of organization of Effective visuals solution. work. work; may have work is evident. and organization script/flow = minor errors or enhance the organization with omissions. solution. graphs, diagrams, tables Verbal Critical Has not yet Includes some Includes concluding Uses concluding Effectively Thinking demonstrated concluding statements and statements and interprets the reflection on statements reflects somewhat effectively answer and reflects reflection on solution or on on the answer and interprets the on both its application and reasonableness its reasonableness. meaning of the reasonableness reflects on of answer. answer and its and how the reasonableness of reasonableness. features of the answer algorithm affect the graph