MATH 107, QUIZ 3 Due Date: Sunday, September 18, 2016 NAME: _______________________________ I have completed this assignment myself, working independently and not consulting anyone except the instructor. INSTRUCTIONS There are 8 questions/problems (on five pages), some with multiple parts. The quiz is worth 100 points, equivalent to 8% of your final course grade. This quiz is open book and open notes. This means that you may refer to your e-text/textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish, provided you turn in your quiz no later than 11:59 PM (US Eastern Time Zone) Sunday, September 18, 2016. Show work/explanation. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also; a single file in pdf format, with your name and quiz number in the file name, is preferred. In your document, be sure to include your name and the assertion of independence of work. If you have any question, please post it in \"Ask the Professor\" discussion on LEO if the answer to your question would benefit others in class; otherwise, please contact me privately via e-mail. PLEASE READ THE QUESTIONS/PROBLEMS CAREFULLY; SHOW ALL YOUR WORK AND REASONING, Just the answer, without supporting work, will receive no credit. 1. [30 points] Draw the graph of a function f(x), with known beginning and end (filled circles, no arrows at the ends) that has the characteristics (in any order that you improvise) listed in Parts (a) through (f) below (please carefully read the entire problem first) and then answer the related questions; finally, provide answers to Parts (g) through (l) that follow. (a) Three increasing segments; write the intervals within which the function is increasing. (b) Two decreasing segments; write the intervals within which the function is decreasing. (c) One local maximum; write the coordinates of the local maximum. (d) One maximum; write down the coordinates of the maximum. (e) One local minimum; write the coordinates of the local minimum. (f) One minimum; write the coordinates of the minimum. (g) Write the interval representing the domain of the function. (h) Write the interval representing the range of the function. (i) Write the zeros of the function f(x), if any, by examining the graph drawn. (j) Find the \"average rate of change\" of f(x) over the interval bounded by the minimum and the maximum of the function. (k) Briefly explain (without any re-graphing) how the graph of f(x 1.5) would look like. (l) Among the Parts (a) through (j) above, choose just one part and provide answers that would be based on f(x 1.5). 1 2. [27 points] Showing your work for all parts of this problem, (a) Graph the function f(x) = x - 1 on a large, clear, and neat coordinate system; label it. (b) What are the slope, the y-intercept (an ordered pair), and the x intercept (an ordered pair) of the line graphed? (c) Showing your work, find g(x) in such a way that its graph would be a vertical (upward) shift of the graph of f(x) by 2 units. Graph g(x) and on the x-y plane and label it. (d) In the list given below, mark the correct statements that apply, if any. Support your answer(s) analytically and briefly. d1 - f(x) and g(x) are reflections (\"mirror images\") of each other about the y-axis. d2 - f(x) and g(x) are reflections (\"mirror images\") of each other about the x-axis. d3 - f(x) and g(x) are reflections (\"mirror images\") of each other about the line y = x. d3 - f(x) and g(x) are reflections (\"mirror images\") of each other \"about the origin.\" d4 - None of the above. (e) What kind of transformation along the x-axis has f(x), the original line, experienced to become g(x)? Support your answer analytically (by resorting to an applicable transformation rule). (f) Subject f(x) to a vertical scaling of your choice (may I suggest a scaling factor of 2). Call the new function h(x); write, graph, and label it. (g) What changes have the points on f(x), the original line, experienced to become h(x)? (HINT: Compare the two lines and state the common change based on the transformation that has taken place.) (h) Subject f(x) to a horizontal scaling of your choice (may I suggest a scaling factor of 2). Call the new function k(x); write, graph, and label it. (i) What changes have the points on f(x), the original line, experienced to become k(x)? (HINT: Compare the two lines and state the common change based on the transformation that has take place.) 3. [10 points] Considering the fact that under specific physical/chemical conditions, water freezes at 32 degrees Fahrenheit (F) on the Fahrenheit temperature scale or at 0 degree Celsius (0) on the Celsius temperature scale, and boils at 212 F or 100 C, showing your work, (a) Find the linear relationship between the Fahrenheit and Celsius temperature scales as F = f(C) and write it in the slope-intercept form. HINT: begin with graphing the freezing and boiling points of water on the C-F plane (similar to x-y plane) and use the slope-point format for the equation of a line [using C and F, instead of x and y] and proceed). (b) If, on the Fahrenheit temperature scale, the normal/natural body temperature of a healthy person is 98.6 F, what would be the body temperature in C of a patient who is running a fever 2 degrees Fahrenheit above normal? (c) At what temperature both temperature scales show the same value? 2 4. [6 points] 5. [6 points] 6. [8 points] Considering the fundamental definition of |x|, and piecewise-defined functions, 7. [4 points] 3 4 8. [9 points] Given the function f(x) = |x| - |x + 2|, (a) Convert f(x) to a piecewise-defined function, based on the fundamental definition of |x|, as explained in Section 2.2 (for Week 3), and the examples in that section. (b) Graph the function. (c) Find the x- and y-intercepts of the graph. (d) Find the zeros of the function. (e) Determine the domain of f(x). (f) Determine the range of f(x). 5