NOTE: Type your write-up, except for math equations (these may be
hand-written). Clearly indicate all answers and staple all work. In Your
Own Words questions must be answered with complete correct sentences.
Graphing Project #1
1. Given the function f(x)=x2, graph each of the following functions and
identify the shift(s) of f(x) = x2 occurring in each function. Describe
each shift that occurs. Hand in these printed out graphs along with the
description of the shifts.
A) y = x2 - 2 Enter as y=x^2-2
B) y = (x + 1)2 Enter as y = (x+1)^2
C) y = -(x + 2)2 + 1 Enter as y = -(x+2)^2 +1
2. Identify the formulas that correspond to each of the following
functions. Note that one of these graphs is y=x2. The other 3
graphs are a shift/stretch/reflection of the function y = x2.
Graph these functions and hand in the printed graphs with the formulas.
3. The following is the graph of y = 2e0.10x where y
represents the price of a $2 item adjusted for 10% inflation after x
years. Note: This is entered in
graphmatica as y =
2*exp(0.10*x) and the graph is zoomed out twice .
Use shift rules to find equations for the graphs shown below:
Hand in the printed out graphs along with the formulas for the three
functions.
4. In Your Own Words: Describe how a function f(x) must be modified in
order to shift its graph up or down, right or left, flip its graph upside
down, or vertically stretch its graph.
NOTE: Type your write-up, except for math equations (these may be
hand-written). Clearly indicate all answers and staple all work. In Your
Own Words questions must be answered with complete correct sentences.
Graphing Project #2
1. Factor and solve y = x4 - 7x2 + 10. Then, graph
this function and use the zoom-in feature to zoom in on each x-intercept
to verify that each zero you found was correct. Remember that if (a,0) is
an x-intercept, then x=a is a zero. Find the value of each zero out to the
3rd decimal place. Hand in the printed graph and all of the work involved
in factoring and solving for the real zeroes.
2. Graph f(x) = x3 + 2x2 - 3 on graphmatica. Use the
zoom-in feature to zoom in on the one real zero. Divide f(x) by (x - r)
where r = the real zero you found and then solve the resulting quadratic
equation to find the other two complex zeroes, and then write f(x) in
factored form. Hand in the graph of f(x), the factored form, and all of
the work involved in dividing by the actual zero to find all three of the
zeros and the factored form.
3. Find a polynomial function that matches the 3rd degree polynomial given
below. Print out and hand in the graph of the function that you found.
Note that this polynomial is "stretched" vertically.
4. Use Graphmatica to find the zeros of y = x3 - 3x - 1 . To do
this, first graph this function, then zoom-in on each x-intercept. Find
each zero to 3 decimal places and hand in each of the 3 zoomed-in graphs.
5. Write the approximate factored form of the function y = x3 -
3x - 1 and hand in. Remember that if x = a is a zero, then (x - a) is a
factor.
6. In Your Own Words: explain why you can always find all the zeros of a
3rd degree polynomial once you find the first zero. In other words, why is
it always the case that once you find the first zero, you can find the
other 2 zeros?
7. A certain bacteria increases in number exponentially according to the
formula y = 2e0.10x where x is the number of days and y is the
population of the bacteria. The graph is shown below - it is entered in
Graphmatica as
y = 2*exp(0.10*x) and the graph is zoomed out twice .
Plot the graph on Graphmatica and use the zoom out and zoom in features to
answer the following: What will the population be on day 25? On about what
day will the population be 50?
NOTE: Type your write-up, except for math equations (these may be
hand-written). Clearly indicate all answers and staple all work. In Your
Own Words questions must be answered with complete correct sentences.
Graphing Project #3
1. Graph the function f(x) = (x^2 + 3x+2)/(x^2-9) on Graphmatica.
Identify the x-intercepts from the graph. Solve the equation above when
f(x) = 0 to verify these x-intercepts.
Identify the vertical asymptotes. Let the denominator = 0 and solve for x
to verify these vertical asymptotes.
Hand in the printed graph and all the work involved in finding the
intercepts and the vertical asymtotes of f(x).
2. Graph the same function from #1 f(x) = (x^2 + 3x+2)/(x^2-9) . Zoom out
3 times. Then graph y=1 over the graph of f(x).
Print out these two functions on one graph. Hand in this printout of the
two graphs.
3. In Your Own Words: Explain why the rule that states that you divide
leading coefficients to get y = 1 as the horizontal asymptote works.
4. Identify and graph a rational function that has vertical asymptotes of
x = 2 and x = -2 and has x-intercepts of (3,0) and (-3,0) and has a
horizontal asymptote of y = 2.
Hand in the graph of the function that you found and the work involved in
finding the equation of the function.
5. Graph f(x) = (x^2-3x-2)/(x+2) .
Identify the vertical asymptote. Long divide the numerator by the
denominator to find the slant asymptote.
Zoom out 4 times. Then graph the equation of the slant asymtote over the
graph of f(x). (Your graph of f(x) should approach the slant asymptotes.)
Hand in the work involved in finding the asymptotes and the zoomed out
graph containing f(x) and its slant asymptote.
QUESTIONS 6-10 ON NEXT PAGE>>>
6. In Your Own Words: Explain why long division of f(x) results in the
slant asymptote.
7. If there is 4% inflation, the cost of an item that currently costs $1
will cost y=1*e0.04x dollars after x years. Graph this function
for x-values ranging from 0 to 60 and hand in this graph. Note: Enter this
as y=1*exp(0.04x). Also, answer the following:
In what year will the cost of the item by $6?
How much will the item cost in x=60 years?
Hand in the graph and the answers to the two questions asked.
8. Repeat Problem 7 except use 8% inflation.
9. In Your Own Words: Does doubling the inflation rate double prices after
50 years? Why or why not?
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