Casio FX-CG10 User manual

Fostering Advanced Algebraic Thinking with Casio Technology
CHAPTER 1—POLYNOMIAL FUNCTIONS
Mathematics curricula often emphasize the practical applications of quadratic
functions, yet there is limited justification provided for studying higher-order
polynomial functions. In this chapter, we do just that!
We motivate the study of higher-order polynomial functions by introducing a classic
investigation into maximizing the volume of a cone—to hold ice cream, of course.
We then progress to using polynomial functions to model trends in data. For
example, a quadratic is used to model the results of a study on the empowerment of
women in Bangladesh, a much different use of the quadratic function than generally
found in algebra curricula. Higher-order polynomial functions are also used to
model the average nightly hotel rates in US cities and the number of lives saved
using airbags, seatbelts, and motorcycle helmets.
Features of polynomial functions, including intervals of increase and decrease and
relative minima and maxima, are the focus of these investigations. The goal is for
students to identify these features, and interpret the features with respect to the
data. We are hopeful that the investigations will help students better understand
polynomial functions and develop a greater understanding of the role these
functions can assume in identifying trends in data.


Chapter 1: Polynomial Functions
Investigation 1.1—Homemade Waffle Cones
Did You Know?
Italo Marchiony sold homemade ice cream from a pushcart on Wall Street. Often,
his customers wandered off with the ice cream dishes, or worse yet, broke them.
To curb these expenses, Italo baked edible waffle cups with sloping sides and flat
bottoms. He patented his idea in 1903.
Reference: http://www.makeicecream.com/hisoficecrea.html
The local scholarship committee is planning to sell ice cream cones at high school
events. The scholarship committee is interested in serving the ice cream in waffle
cones, and, the committee has a waffle maker that will make circular waffles 21
centimeters in diameter.
Let’s investigate how the scholarship committee can design a waffle cone with the
greatest volume using a circular waffle, with a diameter of 21 centimeters. To do so,
cut out a circle (from paper, not waffle yet) that is 21 centimeters in diameter.
Remove a sector from the circle, and construct a cone. Be sure to tape or glue your
cone together, as we will be investigating the dimensions of the cone that will
maximize its volume.
a. What is the slant height of the cone? Did you need to use a ruler to
determine this measurement? Explain.
b. Without measuring, write an equation for the radius of the base of the cone
you constructed in terms of the arc length you removed from the circle.
[Hint: Write an equation for the circumference of the base of the cone first.]
c. Again, without measuring, write an equation for the height of the cone you
constructed in terms of the arc length you removed from the circle.
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Fostering Advanced Algebraic Thinking with Casio Technology
Investigation 1.1—Homemade Waffle Cones (continued)
d. Write an equation for the volume of the cone in terms of the arc length you
removed.
e. Determine the arc length that must be removed to construct a cone with the
maximum volume.
f. What would be the dimensions of the cone that has a maximum volume?
Construct another cone from a circle of diameter 21 cm, but now construct a cone
with maximum volume.
g. Do you recommend the scholarship committee make waffle cones with these
dimensions? Why or why not?


Chapter 1: Polynomial Functions
Sample Solution—Homemade Waffle Cones
a. What is the slant height of the cone? Did you need to use a ruler to
determine this measurement? Explain.
The slant height of the cone is the radius of the circle it was constructed from,
so the slant height is approximately 10.5 centimeters.
b. Without measuring, write an equation for the radius of the base of the cone
you constructed in terms of the arc length you removed from the circle.
[Hint: Write an equation for the circumference of the base of the cone first.]
The circumference of the base of the cone is the circumference of the paper
circle less the arc length removed. Since the diameter of the paper circle is 21
centimeters, the circumference is 21
S centimeters. Therefore, the
circumference of the base of the cone is 21
S centimeters less the length of the
arc removed, a, or 21 S a centimeters.
To find the radius of the base of the cone, we recall the formula for the
circumference of a circle is C = 2
S r. Solving this equation for the radius, r,
gives the circumference divided by 2 S . So, the radius of the base of the cone
is r =
S
S
2
21 a= 10.5 
S 2
a.


Fostering Advanced Algebraic Thinking with Casio Technology

Sample Solution—Homemade Waffle Cones (continued)
c. Again, without measuring, write an equation for the height of the cone you
constructed in terms of the arc length you removed from the circle.
The height of the cone can be determined using the Pythagorean Theorem, as
the radius of the base of the cone and the height of the cone are two legs of a
right triangle in which the slant height is the hypotenuse. Therefore, (radius of
cone)2+ (height of cone)2= (slant height of cone)2, or (height of cone)2=
(slant height of cone)2(radius of cone)2. Substituting the expressions from
parts a and b:
(height of cone)2= (10.5)2
2
2
510 ¸
¹
·
¨
©
§
S
a
.
(height of cone)2= (10.5)2

»
»
¼
º
«
«
¬
ª¸
¹
·
¨
©
§

2
2
2
510
510
SS
a.
.
(height of cone)2= 2
2
4
510
S
S
aa. 
height of cone = 2
2
4
510
S
S
aa. 
Note: we are only considering the positive root.
d. Write an equation for the volume of the cone in terms of the arc length you
removed.
Recall, the volume of a cone is V = hr 2
3
1
S . Using the expressions from parts b
and c, the volume of the cone we constructed is:
V = 2
2
2
4
510
2
510
3
1
S
SS
S
aa.a
.
¸
¹
·
¨
©
§


Chapter 1: Polynomial Functions
e. Determine the arc length that must be removed to construct a cone with the
maximum volume.
Table:
Sample Solution—Homemade Waffle Cones (continued)
xPress pto return to the Main Menu, then 7to access the Table
mode.
xInput the equation for the volume of the cone as a function of the arc
length removed from the circle into Y
Y1
. [Notice: This may be
challenging for students, as they must be very careful to use grouping
symbols to ensure the order of operations are adhered to. Yet, the
natural display of the Casio PRIZM calculator allows the students to
quickly assess whether or not they entered the function correctly.]
xNow, press y(SET) to set the input values for x (a, arc length) to be
used in the table. Let’s create a table of arc lengths from 0 to 75
centimeters, inclusive. To do so, input the following values: S
START:
0l, next to E
END:
75l.
xLet’s set the S
STEP
value to be 5, meaning the x-values in the table will
be increasing by 5. Press 5l.


Fostering Advanced Algebraic Thinking with Casio Technology

From the table, we determined that the arc length to be removed should be
greater than 5 centimeters and less than 15 centimeters. To be more precise,
we either need to change the settings of our table, or use a different
representation—such as the graph or the equation. It is important that
students are able to move flexibly between and within representations, and
that they also learn to identify the strengths and weaknesses of each
representation when solving problems.
Graph:
To construct a graph using the calculator:
xPress pto return to the Main Menu, then 5to access the Graph
mode. Notice the function used in the Table mode appears in the Graph
mode as well.
xSet the view window, by press Le
(V-Window)
. The length of the
arc removed must be greater than or equal to 0. We have chosen to set
the input values for the view-window as 0 to 75. To do so, input the
following values: X
Xmin:
0l; X
Xmax:
75l; s
scale:
5l.
xSimilarly, the volume of the cone must be greater than or equal to 0; so
we have chosen to input the following values: Y
Ymin
:0l;
Xmax:
500l; s
scale:
100l.
xPress dto return to the equation screen, and press u(DRAW) to
draw the graph of the equation in the view window you defined.

Sample Solution—Homemade Waffle Cones (continued)

Chapter 1: Polynomial Functions
Sample Solution—Homemade Waffle Cones (continued)
xTo determine the arc length that will maximize the volume of the cone,
press Ly(
G-Solv)
for the graph solver, then press w(MAX). The
calculator will identify the maximum of the graph selected.
Equation:
To determine the arc length that must be removed to maximize the volume of
the cone symbolically requires knowledge of the concept of derivative. This is
beyond our expectations of students studying advanced algebraic concepts.
Yet, notably, graphing technology often provides students with access to
mathematical concepts that they may not be prepared to investigate
symbolically.


Fostering Advanced Algebraic Thinking with Casio Technology
f. What would be the dimensions of the cone that has a maximum volume?
Since the arc length of the cone, with a maximum volume, is approximately
12.1 cm, we need to substitute this into the expressions from parts b and c to
determine the radius and the height of this cone. Recall, from part a, we know
the slant height is 10.5 centimeters.
We chose to enter the equations from parts b and c into the Table mode of
the calculator to approximate the dimensions of the cone with maximum
volume, as follows:
Sample Solution—Homemade Waffle Cones
The radius and the height of the cone are approximately 8.6 and 6.1
centimeters, respectively.
g. Do you recommend the scholarship committee make waffle cones with these
dimensions? Why or why not?
If the goal is to make waffle cones with the greatest volume, the committee
should make the waffle cones with these dimensions. However, students may
also want to discuss if these dimensions result in a cone that “fits” the human
hand. Some containers are manufactured to be easy for the consumer to
handle or use, and others are manufactured with the goal of being
aesthetically appealing.


Chapter 1: Polynomial Functions
Investigation 1.2—Motorcycles and Helmets
Did You Know?
If a vehicle equipped with an air bag experiences a hard impact, generally in excess of 10
miles per hour, a crash sensor sends an electronic signal that triggers an explosion. The
explosion generates nitrogen gas that inflates the air bags, and they burst from their
locations and inflate in less than one-tenth of a second. The air bag absorbs impact
energy by forcing gas out through a series of ports in the air bag fabric.
Reference: www.britannica.com/EBchecked/topic/44957/automobile
In the United States, each year the National Highway Traffic Safety Administration
(NHTSA) estimates the number of lives saved by different protective devices. The
following functions approximate the number of lives saved each year by airbags,
safety belts, and motorcycle helmets, where t= 0 represents 1990.
Airbags: LA= 0.0444t4+ 0.0362t3+ 22.361t211.147t+ 41.81
Safety-Belts: LS= 0.119t5+ 4.8727t471.739t3+ 443.81t2321.73t+ 6601.5
Motorcycle Helmets: LH= 0.0548t4+ 1.9586t315.87t2+ 34.646t+ 626.26
a. Graph all three functions on the same xy-coordinate plane. For what domain
do you think these functions will provide accurate estimates of the lives saved
as a result of these safety devices?
b. Find the y-intercepts of each function. What is the real-world meaning of the
y-intercepts? Explain.
c. Do any of the functions have x-intercepts? Explain.
d. Which of the protective devices save the most lives each year?


Fostering Advanced Algebraic Thinking with Casio Technology
Investigation 1.2—Motorcycles and Helmets (continued)
e. Which of the protective devices are the most effective in saving lives? How is
this question different from part d, and is there enough information to
determine this?
f. What is the estimated increase in the number of lives saved by each of the
protective devices between 2000 and 2006?
g. What are the real-world meanings of these estimated increases?
h. Write a few statements summarizing the general trends in the number of lives
saved for each of these protective devices.
Reference: US Dept of Transportation, National Highway Traffic Safety Administration.
Traffic Safety Facts 2007. (Washington DC, 2008)
i. The following table includes the number of lives saved by child restraints
from 1995 to 2007. Do the data appear to have a linear trend? What is the
correlation coefficient of a linear regression analysis?
Year
1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
Lives
Saved by
Child
Restraints
in the U.S.
408 480 444 438 447 479 388 383 447 455 424 427 382
Source: US Dept of Transportation, National Highway Traffic Safety Administration.
Traffic Safety Facts 2007. (Washington DC, 2008)
j. Try a quadratic and a cubic regression. Which is the best fit—the linear,
quadratic, or cubic function? Explain.


Chapter 1: Polynomial Functions
Investigation 1.2—Motorcycles and Helmets (continued)
k. Write a few statements to summarize the general trend in the number of lives
saved by child restraints in the United States between 2000 and 2007.


Fostering Advanced Algebraic Thinking with Casio Technology
Sample Solution—Motorcycles and Helmets
a. Graph all three functions on the same xycoordinate plane. For what domain
do you think these functions will provide accurate estimates of the lives saved
as a result of these safety devices?
Graph:
xPress pto return to the Main Menu, then 5to access the Graph
mode.
xEnter the three functions in as Y
Y1
, Y
Y2
, and Y
Y3
.
xSet the view window by pressing Le
(V-Window)
.The time in years
since 1990 must be greater than or equal to 0. We have chosen to set
the input values for the view-window as 0 to 30. To do so, input the
following values: X
Xmin:
0l; X
Xmax:
30l; s
scale:
2l.
xSimilarly, the number of lives saved must be greater than or equal to 0;
so we have chosen to input the following values: Y
Ymin
:0l; Y
Ymax:
20000l; s
scale:
1000l.
xPress dto return to the equation screen, and press u(DRAW) to
draw the graph of the equation in the view window you defined.


Chapter 1: Polynomial Functions
We decided [0, 16], representing 1990 to 2006, is an appropriate domain for
these functions. We determined this by finding the maximum values of the
functions Y
Y1
and Y
Y2
for airbags and seatbelts, respectively.
Sample Solution—Motorcycles and Helmets (continued)
xPress Ly
(G-Solv)
,then press w(Max). Arrow up or down until
the function you are interested in flashes and press l. Below we show
two of these maximum points:
We do not believe the number of lives saved as a result of these safety devices
will decrease in the near future, as we believe these products are continuously
being improved and the amount of people in automobile accidents in the
United States is not declining significantly. Notice, this means we believe the
functions allow us to identify trends between 1990 and 2006, but do not allow
us to extrapolate beyond the domain.
b. Find the y-intercepts of each function. What is the real-world meaning of the
y-intercepts? Explain.
Table:
xPress pto return to the Main Menu and 7to access the Table mode.
xPress y(SET) to set the input values for x (t, years) for the table. Let’s
create a table of the lives saved from 1990 to 2006, inclusive. To do so,
input the following values: S
START:
0land E
END:
16l.


Fostering Advanced Algebraic Thinking with Casio Technology
xLet’s set the S
STEP
value to be 1, meaning the x-values in the table will be
increasing by 1, so press 1l.
Sample Solution—Motorcycles and Helmets (continued)
The y-intercepts are the y-values that correspond to x = 0, as shown here.
Graph:
xPress pto return to the Main Menu, press 5to access the Graph
mode and u(DRAW) to draw the graph.
xTo determine the y-intercept of each function, press Ly(
G-Solv)
and r(Y-ICEPT).
xOne of the graphs will begin flashing. Press lto select this graph, or
arrow down to a different graph and then press lto select that graph
instead.
xThe calculator will then identify the y-intercept of the graph selected.
xRepeat these steps for each of the functions.


Chapter 1: Polynomial Functions
Airbags: Seatbelts:
Sample Solution—Motorcycles and Helmets (continued)
Motorcycle Helmets:
Equation:
To determine the y-intercepts of each of these functions using the equations,
we substituted x = 0 into each function and determined the corresponding y-
value.
L
A= 0.0444(0)4+ 0.0362(0)3+ 22.361(0)211.147(0) + 41.81 = 41.81
LS= 0.119(0)5+ 4.8727(0)471.739(0)3+ 443.81(0)2321.73(0) + 6601.5 =
6601.5
L
H= 0.0548(0)4+ 1.9586(0)315.87(0)2+ 34.646(0) + 626.26 = 626.26
Using the table, the graph, and the equations, we identified the y-intercepts of
the three functions to be approximately 41.81, 6601.5, and 626.26. This means
there were approximately 42, 6602, and 626 lives saved in the year 1990 in the
United States by airbags, seatbelts, and motorcycle helmets, respectively.


Fostering Advanced Algebraic Thinking with Casio Technology
c. Do any of the functions have x-intercepts? Explain.
Yes, but we don’t believe these have any real-world meaning. The x-intercepts
indicate the years in which these safety devices would save zero lives. Until
there are no people driving in the United States, we don’t think this will ever
be the case.
Sample Solution—Motorcycles and Helmets (continued)
d. Which of the protective devices save the most lives each year?
The seatbelts save the most lives per year.
e. Which of the protective devices are the most effective in saving lives? How is
this question different from part d, and is there enough information to
determine this?
Clearly, seatbelts save the most lives, but is that because more drivers wear
seatbelts than have airbags or wear motorcycle helmets. To determine which
of the protective devices is the most effective, we would like to compute the
ratio of lives saved to lives lost while using each of the protective devices.
Unfortunately, we do not have enough information to do so.
f. What is the estimated increase in the number of lives saved by each of the
protective devices between 2000 and 2006?
Since x = 0 represents 1990, x = 10 and x = 16 represent 2000 and 2006,
respectively.
Table:
xPress pto return to the Main Menu, press 7to access the Table
mode, then press u(TABLE) to create the table.


Chapter 1: Polynomial Functions
Sample Solution—Motorcycles and Helmets (continued)
xScroll through the table to find the number of lives predicted to have
been saved by each protective device in 2000 and 2006.
Graph:
xPress pto return to the Main Menu, press 5to access the Graph
mode, then press u(DRAW) to view the graph.
xTo determine the y-value of each function when x = 10 and x = 16 both,
press Ly(
G-Solv)
, ufor more options, and q(Y-Cal) to calculate
a y-value.
xOne of the graphs will begin flashing. Press lto select this graph, or
arrow down to a different graph and then press lto select that graph
instead.
xInput 10lfor the x-value and the calculator will then identify the
corresponding y-value of the graph selected. Repeat this for x = 16.
xNow, repeat all of these steps for the other two functions.
A
Airbags:


Fostering Advanced Algebraic Thinking with Casio Technology
Sample Solution—Motorcycles and Helmets (continued)
S
Seatbelts:
M
Motorcycle Helmets:
Equation:
To compute the average increase in lives saved from 2000 to 2006, let’s
determine how many lives were saved by each of these safety devices in both
2000 and 2006. To do so, substitute x = 10 and x = 16 into each of the
functions, and calculate the corresponding y-values.

Airbags:
L
A= 0.0444t4+ 0.0362t3+ 22.361t211.147t+ 41.81
L
A(10) = 0.0444(10)4+ 0.0362(10)3+ 22.361(10)211.147(10) + 41.81 |1,759
LA(16) = 0.0444(16)4+ 0.0362(16)3+ 22.361(16)211.147(16) + 41.81 |2,826
