I really need help with these questions, can you help me? 

 

 

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). 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.
1 (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? 4. [6 points] 2 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