1. Draw the graph of a function y = g(x) such that (a) it has an “open” (exclusive) beginning somewhere in the left side of the y-axis and has a “closed” (inclusive) ending somewhere in the right side of the y-axis, with” one “valley” and one “peak,” in any order, and three x-intercepts, not passing through the origin. It should be noted that the graph asked for can be any graph drawn by hand to meet the general requirements stated. (b) Identify the informative points on the graph by showing each as an ordered pair on the graph. (c) Write “y = g(x)” somewhere next to its graph for identification. (d) Find the zeros of the function drawn and write them using correct mathematical notations. (e) Find the x-intercepts of the graph of g(x) and write each as an ordered pair. (f) State the relationship between zeros of g(x) and the x-intercepts of the graph of g(x). (g) Find the y-intercept of the graph of g(x) and write it as an ordered pair. (h) Write the mathematical interval representing the domain of the function drawn. (i) Write the mathematical interval representing the range of the function drawn. (j) State, in mathematical notations (interval(s)), where the function would be decreasing. (k) State, in mathematical notations (interval(s)), where the function would be increasing. (l) Write the coordinates of the absolute minimum of the function, if any. (m) Write the coordinates of the absolute maximum of the function, if any. (n) Write the coordinates of the relative minimum of the function, if any. (o) Write the coordinates of the relative maximum of the function, if any. (p) If y = h(x) is a transformation of y = g(x), such that h(x) = 0.5 g(x-1), draw the graph of h(x), preferably on the same coordinate system (x-y plane) that you used for the graph of g(x), or separately. You may use a “thicker pen” or simply use dashes (instead of solid line) while drawing h(x). Write “y = h(x)” somewhere next to its graph for identification. Identify the coordinates of the informative points on h(x). (q) What is the transformation that the function g(x) went through to become h(x) called? (r) Find the “average rate of change” of g(x) over the closed interval bounded by the relative minimum and the relative maximum of the function. Hint: See Definition 2.3 of the eBook. (s) FOR EXTRA CREDIT (2 points) Draw y = |g(x)| and identify the new points on it. 2. Linear relationship/Mathematical Modeling. 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) Graph the freezing and boiling points of water on “C-F plane,” in place of an “x-y plane.” (b) Find the slope of the line F = f(C), passing through the freezing point and the boiling point provided above. Keep working with simplified fractions, not decimals. (c) Briefly explain what the interpretation of your finding for part (b) above is. Give a simple example. (d) Derive the linear relationship between the Fahrenheit and Celsius temperature scales in slopeintercept form as F = mC + b. (e) Showing your work, check the function/equation that you derived for Part (d) above, using a point residing on the line. (f) Derive (set up) the linear relationship between the Fahrenheit and Celsius temperature scales in point-slope form, keeping in mind that C and F have replaced x and y, respectively. (g) Based on the linear relationship found for Part (d) or for Part (f) above, 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 2.1 F above normal? Report your answer to the tenth of a degree. (h) At what temperature both temperature scales show the same value? (Hint: F = C, then, replace C with F (or vice versa) in one of the C-F relationships that you found above; then solve the equation for the unknown.) 3. [8 points] Mathematical modeling The width a rectangular tablecloth is 5 2/5 (five and two fifths) feet shorter than its length. If the tablecloth is W feet wide and L feet long, (a) Express the perimeter of the tablecloth (P) as a function of its length, i.e., find P = f(L). NOTE: In developing your mathematical model (the function), you are not to work with specific numerical values for the length and the width. Work with L, W, and their relationship, knowing that the perimeter of a rectangle is the sum of its four sides. (b) Using your mathematical model (function) developed in Part (a), determine the perimeter of the tablecloth when its width is 6 3/5 (six and three fifths) feet. Report the answer to two decimal places. Do not forget the unit for your answer. 4. In the x-y plane, can a function, say y = f(x), be symmetric about the x-axis? Support your answer with brief, but mathematically convincing, explanation/example. Read the question carefully. 5. Considering the fundamental definition of |x|, provided in Section 2.2, and piecewise-defined functions, Graph the following function, using the x-y coordinates provided or the ones that you would draw: |x|, for x < – 3 f(x) = x2/5, for – 3 ≤ x ≤ 2 – 2x, for x > 2 6. [20 points] Given f(x) = – |x – 4|, (a) Find the zeros of the function. (b) Find the x- intercept(s). (c) Find the y-intercept. (d) Rewrite the function without absolute values (as a piecewise function). (e) Graph the function. (f) Determine the domain and the range of the function. (g) State in interval format where the function is decreasing or increasing. (h) Determine the relative and absolute extrema. 7. (a) Find the graph of g(x) following the information provided in the graph of f(x) below (next page). (b) Support your answer briefly by stating what transformation(s) the function f(x) has gone through to become g(x). (c) Having found g(x), write the domain and the range of g(x) in interval format.
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