Science:Math Exam Resources/Courses/MATH104/December 2011/Question 02 (g)

MATH104 December 2011

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Question 02 (g)
Draw the graph of ƒ(x) on the graph provided. Accurately place all critical points and inflection points, indicate all asymptotes, and make sure your graph correctly shows where ƒ(x) is increasing and decreasing and correctly shows its concavity.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
You don't even need to use the expression for $f(x)$; the answers from parts a - f should be sufficient to draw the graph.

Hint 2
Try going step by step. Draw the asymptotes (part f). Draw the local maxima and minima (part e). Draw the inflection point (part e). Look at where the function is increasing/decreasing (part c) and try to connect the points you've already drawn.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Following the order given in Hint 2, we first draw the asymptotes. We then draw the maxima and minima. We draw the inflection point. Finally, we use information about where the function is increasing and decreasing, as well as its concavity, to connect these points and finish the graph. A graphing calculator would produce the following picture:

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