Usefulness of Graphs Questions
Order ID: 89JHGSJE83839 Style: APA/MLA/Harvard/Chicago Pages: 5-10 Instructions:
) This exercise illustrates limitations to the usefulness of graphs. Consider the function
f: R ? R,
f (x) = {?(e^((-1)?x^2 )&x?0@0&x=0)?
Why is it reasonable to de?ne f (0) = 0?
Using technology, construct a table of values, for the inputs {0, 0.1, 0.2, 0.3, 0.4}.
Using your answer to (b), attempt to construct a rough graph of f (x) over the interval
[0, 0.4].
What property of the function makes it so dif?cult to construct a meaningful graph?
Use technology (graphing calculator or computer algebra system) to graph f. Discuss your results.2)
Let f be a function. Find a formula for the function whose graph is obtained from the graph of f by shifting two units to the left and then by stretching vertically by a factor of three. Then, ?nd the function whose graph is obtained from the graph of f by stretching vertically by a factor of three and then by shifting two units to the left.
George and Laura are trying to graph 2sin(3x). Laura wants to do the horizontal compression followed by the vertical stretch, but George insists on the reverse order. Which method is correct? Why?
John wants to move up, and Teresa is willing to move to the left. Can they graph
sin??(x+?/4)+1?? What two graphing transformations would they use? Does the order of
the transformations matter?
State a general principle about transformations, on the basis of parts (a), (b), and (c).
3) Suppose that the x -intercepts of the graph of f are {4,7}. Find the x -intercepts of the graphs of the following functions. (If it is impossible to tell from the given information, explain why.)
(a) f (x ? 3)
(b) 2 f (7x + 1)
(c) 7 f (x) + 14) Let A be the set of all humans, and let B be the set of all human males. De?ne f: A ? B by the rule f (a) = a s biological father.
(a) In words, describe the range of f.
(b) If y ? B, then (in words) what is the set f -1 [{y}]?
(c) If x ? A, then (in words) what is the set f^(-1)[ f [{x}]]?
(d) If y ? B, then what is the set f [ f^(-1) [{y}]]. Your answer should depend on whether y is in the range of f.
5) Let f: A ? B be a function.
(a) Prove that if S ? A, then f^(-1) [ f [S]] ? S.
(b) Prove that if T ? B, then f [ f^(-1) [T]] ? T.
6) Let f : R? R, f(x) = x^2 (x-3).
Given a real number b, ?nd the number of elements in f^(-1) [{ b }]. (The answer will depend on b. It will be helpful to draw a rough graph of f, and you probably will need ideas from calculus to complete this exercise.)
Find three intervals whose union is R, such that f is injective as a function on each interval.
Use your answers to (a) and (b) to de?ne the domain and codomain of three bijective function7) Let F: R ? R, F (x) = 3 x + 5. We can write F easily as the composite of two functions. Namely, if we de?ne f and g (both with domain and codomain R ) by the rules f ( x ) = x + 5 and g ( x ) = 3 x , then F = f ? g . Find four other ways to write F as the composite of two functions (with domain and codomain R ).
RUBRIC
Excellent Quality
95-100%
Introduction 45-41 points
The background and significance of the problem and a clear statement of the research purpose is provided. The search history is mentioned.
Literature Support
91-84 points
The background and significance of the problem and a clear statement of the research purpose is provided. The search history is mentioned.
Methodology
58-53 points
Content is well-organized with headings for each slide and bulleted lists to group related material as needed. Use of font, color, graphics, effects, etc. to enhance readability and presentation content is excellent. Length requirements of 10 slides/pages or less is met.
Average Score
50-85%
40-38 points
More depth/detail for the background and significance is needed, or the research detail is not clear. No search history information is provided.
83-76 points
Review of relevant theoretical literature is evident, but there is little integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are included. Summary of information presented is included. Conclusion may not contain a biblical integration.
52-49 points
Content is somewhat organized, but no structure is apparent. The use of font, color, graphics, effects, etc. is occasionally detracting to the presentation content. Length requirements may not be met.
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0-45%
37-1 points
The background and/or significance are missing. No search history information is provided.
75-1 points
Review of relevant theoretical literature is evident, but there is no integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are not included in the summary of information presented. Conclusion does not contain a biblical integration.
48-1 points
There is no clear or logical organizational structure. No logical sequence is apparent. The use of font, color, graphics, effects etc. is often detracting to the presentation content. Length requirements may not be met
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Usefulness of Graphs Questions