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Casio fx-CG10 Advanced Operation Manual
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Casio fx-CG10 Advanced Operation Manual

Casio fx-CG10 Advanced Operation Manual

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  • Page 2 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.
  • Page 3 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.
  • Page 4 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.
  • Page 5 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.
  • Page 6 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.
  • Page 7 Yet, the natural display of the Casio PRIZM calculator allows the students to quickly assess whether or not they entered the function correctly.] Now, press (SET) to set the input values for x (a, arc length) to be used in the table.
  • Page 8 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Homemade Waffle Cones (continued) 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.
  • Page 9 Chapter 1: Polynomial Functions Sample Solution—Homemade Waffle Cones (continued) To determine the arc length that will maximize the volume of the cone, ( G-Solv) press for the graph solver, then press (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.
  • Page 10 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Homemade Waffle Cones 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.
  • Page 11 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.
  • Page 12 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.
  • Page 13 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.
  • Page 14 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?
  • Page 15 Chapter 1: Polynomial Functions Sample Solution—Motorcycles and Helmets (continued) 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 Y 1 and Y Y 2 for airbags and seatbelts, respectively. Press (G-Solv), then press (Max).
  • Page 16 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Motorcycles and Helmets (continued) Let’s set the S S TEP value to be 1, meaning the x-values in the table will be increasing by 1, so press The y-intercepts are the y-values that correspond to x = 0, as shown here.
  • Page 17 Chapter 1: Polynomial Functions Sample Solution—Motorcycles and Helmets (continued) Airbags: Seatbelts: 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. = 0.0444(0) + 0.0362(0) + 22.361(0)
  • Page 18 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Motorcycles and Helmets (continued) 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...
  • Page 19 Chapter 1: Polynomial Functions Sample Solution—Motorcycles and Helmets (continued) Scroll through the table to find the number of lives predicted to have been saved by each protective device in 2000 and 2006. Graph: Press to return to the Main Menu, press to access the Graph mode, then press (DRAW) to view the graph.
  • Page 20 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Motorcycles and Helmets (continued) S S eatbelts: M M otorcycle 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.
  • Page 21 Chapter 1: Polynomial Functions Sample Solution—Motorcycles and Helmets (continued) Safety-Belts: = 0.119t + 4.8727t 71.739t + 443.81t 321.73t + 6601.5 = 0.119(10) + 4.8727(10) 71.739(10) + 443.81(10) 321.73(10) + 6601.5 12,853 = 0.119(16) + 4.8727(16) 71.739(16) + 443.81(16) 321.73(16) + 6601.5 15,783 Motorcycle Helmets: = 0.0548t...
  • Page 22 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Motorcycles and Helmets (continued) h. Write a few statements summarizing the general trends in the number of lives saved for each of these protective devices since 1990. Answers may vary, but students should use mathematics to support their statements.
  • Page 23 Chapter 1: Polynomial Functions Sample Solution—Motorcycles and Helmets (continued) Arrow down to X X List and press (LIST) . Arrow down to Y Y List and press (LIST) . This tells the calculator that the x-values for your scatterplot can be found in L L ist 1, and the y-values can be found in List 2.
  • Page 24 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Motorcycles and Helmets (continued) j. Try a quadratic and a cubic regression instead. Which is the best fit—the linear, quadratic, or cubic function? Explain. To perform a quadratic regression: From the scatterplot, press (CALC) and ).
  • Page 25 Chapter 1: Polynomial Functions Sample Solution—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 1995 and 2007. Answers may vary, but students should use mathematics to support their statements.
  • Page 26 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 1.3—Empowerment of Bangladesh Women Did You Know? Approximately 70 percent of the 1.3 billion people in the world that live in poverty are female. Women earn only 10 percent of the world’s income, and they own less than 1 percent of the world’s property.
  • Page 27 Chapter 1: Polynomial Functions Investigation 1.3—Empowerment of Bangladesh Women (continued) Source:www.newparadigmjournal.com/Oct2008/mathematical.htm a. Construct a scatterplot. Use the median age of each interval paired with the mean score of the Women Empowerment Index (WEI). For example, graph the ordered pairs (17, 0.5095) and (22, 0.6615). b.
  • Page 28 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Empowerment of Bangladesh Women (continued) a. Construct a scatterplot. Use the median age of each interval paired with the mean score of the Women Empowerment Index (WEI). For example, graph the ordered pairs (17, 0.5095) and (22, 0.6615).
  • Page 29 Chapter 1: Polynomial Functions Sample Solution—Empowerment of Bangladesh Women (continued) b. Do the data appear to have a linear trend? After performing a linear regression of the data, what is the correlation coefficient of your analysis? The data appear to possibly have a linear trend, although it does not appear to be a strong correlation.
  • Page 30 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Empowerment of Bangladesh Women (continued) The correlation coefficient, r, is approximately 0.70. This suggests the data do have a relatively strong linear trend. Let’s try other types of regression too. c. Now, try a quadratic regression.
  • Page 31 Chapter 1: Polynomial Functions Sample Solution—Empowerment of Bangladesh Women (continued) d. Are all women in Bangladesh equally empowered? Explain. No, not all women in Bangladesh are equally empowered. It appears that women become more empowered as they get older, but then they again become less empowered...
  • Page 32 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Empowerment of Bangladesh Women (continued) Let’s set the S S TEP value to be 5, meaning the x-values in the table will be increasing by 5, so press Press to exit the TABLE Setting mode, and press (TABLE) to create the table.
  • Page 33 Chapter 1: Polynomial Functions Sample Solution—Empowerment of Bangladesh Women (continued) Set the view window by pressing (V-Window). The women’s ages must be greater than or equal to 0, so we have chosen to set the input values for the view-window as 0 to 85. To do so, input the following values: X X min: ;...
  • Page 34 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 1.4—Douglas Fir Trees Did You Know? The modern Christmas tree originated in western Germany. The Germans set up an adorned fir tree in their homes on December 24, the religious feast day of Adam and Eve.
  • Page 35 Chapter 1: Polynomial Functions Investigation 1.4—Douglas Fir Trees (continued) Circumference Height Tree (meters) (meters) 4.97 70.33 6.65 72.92 4.93 65.76 0.29 5.39 0.19 5.12 0.45 8.83 0.62 9.76 0.64 12.08 0.18 5.19 0.55 13.5 10.9 0.39 6.79 0.62 10.66 0.27 11.71 0.41 10.5...
  • Page 36 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 1.4—Douglas Fir Trees (continued) c. Now, try a quadratic regression and a power regression. d. Which is the best fit—a linear, quadratic, or power function? Explain your reasoning. e. What is the equation of the function that best models the data? f.
  • Page 37 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees a. Construct a scatterplot of the circumferences and heights of the trees. To create a scatterplot using the calculator: Press to return to the Main Menu and to access the Statistics mode. First, encourage the students to label each column.
  • Page 38 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) b. Do the data appear to have a linear trend? After performing a linear regression of the data, what is the correlation coefficient of your analysis? Though the data show an increasing function, the trend does not appear to be linear.
  • Page 39 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) The correlation coefficient, r, is approximately 0.95. This indicates that the data have a strong linear trend. Yet, when we visually examine the regression line, the linear equation does not appear to model the data well. Notice, as the circumference of the trees increase, the height of the trees does not seem to continue to increase at the same rate.
  • Page 40 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) To perform a power regression: From the scatterplot, press (CALC), for more options, then (Power). Press (DRAW) to draw the regression equation on the same axes as the scatterplot.
  • Page 41 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) e. What is the equation of the function that best models this data? By comparing the r values, we determined the quadratic regression equation 1.061x + 17.971x + 2.251 models the data most accurately. f.
  • Page 42 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) Let’s set the S S TEP value to be 1, meaning the x-values in the table will be increasing by 1, so press Press to exit the TABLE Setting mode, and press...
  • Page 43 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) Graph: Press to return to the Main Menu and to access the Graph mode Notice the regression equation copied to Y Y = appears in the Graph mode as well. view window pressing (V-Window).
  • Page 44 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) To estimate the height of Douglas fir trees with circumferences of 2, 8, and 12 meters: Press (G-Solv), the graph solver, then press for more options. Press (Y-CAL) for y-calculate. Input for the circumference, or the x-value, and the corresponding y-value will be calculated.
  • Page 45 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) Using the regression equation, we estimate that Douglas fir trees with circumferences of 2, 8, and 12 meters will be 33.9, 78.1, and 65.1 meters tall, respectively. [Note: The approximation of 65.1 meters for a tree that has a circumference of 12 meters should be alarming, since it’s shorter than a tree with a circumference of 8 meters.] Equation:...
  • Page 46 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) Using the r values, the quadratic function was chosen to model the data. However, as the circumference of a tree increases, it would seem the height of the tree should also increase.
  • Page 47 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) Recall, to copy the power regression equation into the Y Y = prompt: Perform the power regression again. Press (COPY) to copy the regression equation to the Y Y = prompt. Table: Students can choose to scroll through the table of the power regression equation to approximate the heights of a Douglas fir tree is 2, 8, and 12...
  • Page 48 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) A graph or the equations could also be used, but the steps will not be outlined here. h. What is the approximate circumference of a tree that is 3 meters tall? 45...
  • Page 49 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) Graph: Press to return to the Main Menu and to access the Graph mode. To determine the circumference of Douglas fir trees that are 3, 45, and 150 meters tall, press (G-Solv) for the graph solver, then press for more options.
  • Page 50 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Douglas Fir Trees (continued) Equation: 0.7823 The power regression equation is y 18.3670x , where x represents the circumference and y represents the height of Douglas fir trees. To predict the circumference of trees 3, 45, and 150 meters tall, we evaluated x when y is equal to these values.
  • Page 51 Chapter 1: Polynomial Functions Sample Solution—Douglas Fir Trees (continued) Screenshots showing these computations, as completed in the Run mode, are shown here.
  • Page 52 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 1.5—Trends in Nightly Rates for Hotels Did You Know? Up on the Saanerslochgrat, in central Switzerland, winter enthusiasts can enjoy nightly accommodations in a unique hotel—an igloo. The Igloo Village is made entirely of snow and is located at an altitude of 1937 meters.
  • Page 53 Chapter 1: Polynomial Functions Investigation 1.5—Trends in Nightly Rates for Hotels (continued) a. Construct a scatterplot of the average nightly rate in hotels for each city as a function of the year. b. Do the data appear to have a linear trend? After performing a linear regression of the data, what is the correlation coefficient of your analysis for each city?
  • Page 54 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels a. Construct a scatterplot of the year and the average nightly rate in hotels for each city. To create a scatterplot for each city using the calculator:...
  • Page 55 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) Press (GRAPH1) to create the scatterplot for Baltimore. Baltimore, MD Graph the data for Charlotte as ordered pairs by pressing (SET), and then (GRAPH2) to set up Graph 2 Repeat the steps listed above, but be sure to indicate that the y-values for the Charlotte scatterplot are in L L ist 3.
  • Page 56 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) Milwaukee, WI b. Do the data appear to have a linear trend? After performing a linear regression of the data, what is the correlation coefficient of your analysis for each city? From a visual inspection, the data do not appear to have linear trends.
  • Page 57 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) Repeat these steps to generate linear regression equations for both Charlotte and Milwaukee. Charlotte, NC Milwaukee, WI The correlation coefficients for Baltimore, Charlotte, and Milwaukee are approximately 0.83, 0.84, and 0.92, respectively. These values suggest a relatively strong linear trend for each city, but visually the linear equations do not seem to model the data well.
  • Page 58 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) c. Now, try a quadratic, a cubic regression, and a quartic regression. To perform a quadratic regression: From the scatterplot, press (CALC), then ). The variable x in a quadratic equation is of degree 2, and for this reason, we press x perform a quadratic regression.
  • Page 59 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) To perform a cubic regression: From the scatterplot, press (CALC), then press ). The variable x in a cubic equation is of degree 3, and for this reason, we press x perform a cubic regression.
  • Page 60 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) To perform a quartic regression: From the scatter plot, press (CALC), for more options, then ). The variable x in a cubic equation is of degree 4, and for this reason, we press x to perform a quartic regression.
  • Page 61 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) d. Which is the best fit—a linear, quadratic, cubic, or quartic function? Explain your reasoning. The linear, quadratic, cubic, and quartic regressions all provide an r value, so let’s compare those values for each city.
  • Page 62 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) Baltimore, MD Charlotte, NC Milwaukee, WI f. Compare the trends in average nightly hotel rates in all of the cities. What trends do you observe? Reviewing the tables and the graphs of the nightly hotel rates, we observe some trends common to all cities.
  • Page 63 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) g. For each city, determine in what years the average nightly rate of hotels was increasing and in what years was it decreasing? To use the regression equation to analyze trends of the nightly hotel rates as a function of time in years, we first need to copy the equations into the Y Y = prompts.
  • Page 64 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) Press (SET) to set the input values for x (years) to be used in the table. Let’s create a table of the nightly hotel rates from 2000 to 2020.
  • Page 65 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) In Charlotte, the nightly price of a hotel decreased from 2000 to 2003, but ever so slightly. The nightly price of a hotel in Charlotte also increased from 2003 to 2008, when it reached a maximum, and then began to decline in 2009.
  • Page 66 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) Set the view window by pressing (V-Window). Let’s graph the average nightly hotel rates for each city as a function of the year from 2000 to 2020. To do this, we have chosen to set the input values for the view-window as 0 to 20.
  • Page 67 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) One of the graphs will begin flashing. Press to select this graph, or arrow down to a different graph and then press to select that graph instead. The calculator will then compute the minimum of the graph selected. Press (G-Solv), then (MAX) to find the maximum average...
  • Page 68 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) From the graph of average nightly hotel rates of Baltimore, as a function of time in years since 2000, it appears that the rates were increasing between 2000 and 2001, and between 2002 and 2007.
  • Page 69 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) From the graph of average nightly hotel rates of Milwaukee, as a function of time in years since 2000, it appears as if the rates were increasing in 2000, but then decreased until 2003.
  • Page 70 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) h. Review the events that occurred in the United States between 2000 and 2009. Why do you think the hotel rates were increasing and decreasing in the years you identified? Explain.
  • Page 71 Chapter 1: Polynomial Functions Sample Solution—Trends in Nightly Rates for Hotels (continued) i. Using the equations that model the data, predict the average nightly hotel rate for 2012 for each city. Graph: To estimate the average nightly hotel rate of each city in 2012: From the graph, press (G-Solv), the graph solver.
  • Page 72 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Trends in Nightly Rates for Hotels (continued) j. How accurate do you think these predictions will be? Explain. These predictions are not accurate. The average nightly hotel rate cannot be a negative number. This is an example of a set of regression equations that provide us a lot of information about the trends within the given data, but are not useful for extrapolating data—making predictions outside of the data set.
  • Page 73 Chapter 1: Polynomial Functions Investigation 1.6—Thrill of the Ride Gravity powers most of a ride on a traditional roller coaster. A large initial climb usually introduces the potential energy for the entire ride. This is then converted to kinetic energy on the first, and generally the sharpest, drop. Thrill seekers are entertained by the velocity of the descent, the inverted loops, barrel rolls, and banked turns.
  • Page 74 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Thrill of the Ride Before determining our model, we will set the picture of the roller coaster as the background on the calculator. In the Statistics mode, press (SET UP). Then, Set the W W indow to Manual.
  • Page 75 Chapter 1: Polynomial Functions Sample Solution—Thrill of the Ride (continued) After identifying all of the points for your quartic function, press and then enter your x-values in L L ist 1 and your y-values in L L ist 2. (Be sure to overwrite the 0 in both lists.) We shown the data points we identified, excluding our final point, which is identified on the previous screen shot.
  • Page 76 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Thrill of the Ride (continued) To identify 2 lists, press . To identify List 1 as the base list, pres To identify List 2 as the accompanying (second list), press . Your data are now sorted from lowest x-value to highest x-value.
  • Page 77 Chapter 1: Polynomial Functions Sample Solution—Thrill of the Ride (continued) Before we continue, a caveat is in order here. We do not know that the design of this particular roller coaster was set up as a quartic model. Furthermore, as we are about to determine the steepest part, we are assuming that we have a reasonable perspective, that this section of the ride is approximately horizontal, to our vantage point.
  • Page 78 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Thrill of the Ride (continued) To find the steepest part, we could find the maximum value now in L L ist 3; however all positive slopes are necessarily larger than any negative ones, which indicate downhill slopes.
  • Page 79 Chapter 1: Polynomial Functions Sample Solution—Thrill of the Ride (continued) We note the minimum, 1.499, is larger in absolute value than the max (0.8888), so the steepest part of the section of roller coaster is, as we had thought, on the downhill. By scrolling through L L ist 3, we find the value of - 1.499 in the 10 and 11 positions.
  • Page 80 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Thrill of the Ride (continued) The calculator allows you to enter a function which describes the slope at each point of the curve. This function is called the derivative. Similar to delta y...
  • Page 81 Chapter 1: Polynomial Functions Sample Solution—Thrill of the Ride We find our minimum point has an x-value of approximately 4.24; at this point, the slope of the quartic curve is approximately 1.57, indicating a drop of 1.57 units per increase of 1 unit horizontally. This point is actually beyond where we can see the roller coaster track clearly and suggests that the steepest part of the graph is actually not between the 10 and 11...
  • Page 82 Fostering Advanced Algebraic Thinking with Casio Technology CHAPTER 6: CONIC SECTIONS Conic sections are formed by the intersection of a plane and a cone. As shown below, four conic sections are possible. Retrievedlfrom http://www.newworldencyclopedia.org/entry/Image:Conic_sections_2.png, August 2010 Conic sections have many interesting and useful applications, from engineering to astronomy, from solar energy to tunnels.
  • Page 83 Chapter 6: Conic Sections Investigation 6.1—Throwing High Did You Know? Terminal velocity is when a falling object stops accelerating. The net force of air resistance and gravity is equal to zero. When the object reaches terminal velocity, it will continue to fall but at a constant rate. If an object is more compact and more dense, it has a greater terminal velocity.
  • Page 84 Fostering Advanced Algebraic Thinking with Casio Technology a. Determine how high up the ball will go. Explain your result. b. Determine when the ball will hit the ground. Explain your result. c. Explore what happens to the maximum height and how long the ball is in the air as the initial vertical speed changes, from 0 feet per second to 50 feet per second, in increments of 5 feet per second.
  • Page 85 Chapter 6: Conic Sections Sample Solution—Throwing High a. Determine how high up the ball will go. Explain your result. Using our equation, , from the GRAPH menu, input the function then press (DRAW) to view the graph. After exploring the function, change the view window settings by selecting (V-Window) setting X X min: 0, X X max: 4, and a s s cale of 1.
  • Page 86 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Throwing High (continued) This tells us that the ball will hit the ground approximately 3.24 seconds after being released. Keep in mind that the graph does not describe the path of the ball, but the height of the ball over time.
  • Page 87 Chapter 6: Conic Sections Sample Solution—Throwing High (continued) Now, press and watch what happens as we incrementally modify B, 5 units at a time; see below for B = 40. The previous graph is barely visible when B = 35. As we incrementally modify B, in a positive direction, we observe how the ball goes higher (the highest point goes up) and stays in the air longer (the x- intercept moves to the right).
  • Page 88 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 6.2—Long-Distance Radio Navigation Did You Know? GPS stands for Global Positioning System. It is made up of three parts; satellites orbiting the Earth, monitoring stations on Earth, and the individual receivers owned by consumers. GPS is used for disaster relief and emergency services, as well as providing driving directions, accurate timing for things such as banking, and even controlling power grids.
  • Page 89 Chapter 6: Conic Sections Investigation 6.2—Long-Distance Radio Navigation (continued) The general form for a hyperbola, with center at the origin and that opens right and left is, , where A represents the distance from the center to a vertex and represents the slope of the asymptotes.
  • Page 90 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Long-Distance Radio Navigation a. How would you orient a coordinate plane, which would be useful in finding the equation of a hyperbola, on which the ship must be located? We can establish a coordinate system so the origin is halfway between the two transmitting stations.
  • Page 91 Chapter 6: Conic Sections Sample Solution—Long-Distance Radio Navigation (continued) d. Assume that A remains at 150; explore what happens to the hyperbola as the value for B changes from 20 to 200 in increments of 20. While viewing the graph, press to return to the input screen and change B to 20.
  • Page 92 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 6.3—Geology and Earthquakes Did You Know? Earthquakes occur when pieces of the earth’s surface called “tectonic plates” bump into each other; the places where they meet are called “fault lines”. There is a fault line that stretches around the edges of the Pacific Ocean and it is called the “Ring of Fire”.
  • Page 93 Chapter 6: Conic Sections Station Station Southern...
  • Page 94 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Geology and Earthquakes a. Suppose one station determines that the epicenter of an earthquake is about 85 miles from the station. Find an equation for the possible location of the epicenter. We will establish a coordinate system with this station at the origin. The epicenter could be anyplace on the circle with the equation .
  • Page 95 Chapter 6: Conic Sections Sample Solution—Geology and Earthquakes (continued) See below for how we inputted the two equations. This screen shot was taken prior to pressing for Y Y 2: so the entire line would be visible. Press (V-Window) to choose an appropriate viewing window. A square window, using a y-base from 0 to 100, was used to display the graph below.
  • Page 96 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 6.4—Extracorporeal Shockwave Lithotripsy Did You Know? The kidneys reabsorb and redistribute 99 percent of the blood volume in the body; and, kidneys have greater blood flow than the brain, heart, or liver. Each kidney is about 4.5 inches long and weighs between 4 and 6 ounces.
  • Page 97 Chapter 6: Conic Sections Sample Solution—Extracorporeal Shockwave Lithotripsy (continued) a. Suppose the half ellipse is 106 units long with a height of 28 units. Orient this ellipse with the center about the origin and determine its equation. One standard form for the equation of an ellipse is: where (H,K) is the center, A is the distance from the center to the vertices along the major axis, and B is the distance from the center to the ends of the minor axis.
  • Page 98 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Extracorporeal Shockwave Lithotripsy (continued) With the graph displayed, press (G-Solv) and then (FOCUS). Press to find the other focus. The stone should be located at one focus, say ( 45, 0), and the source of the shockwaves at the other, (45, 0).
  • Page 99 Chapter 6: Conic Sections Sample Solution—Extracorporeal Shockwave Lithotripsy (continued) The location of the shock waves would be an additional 45 cm to the right of the center, or (87, 7). Below left shows the focus where the source of the shock wave should be placed and below right shows the location of the stone.
  • Page 100 Fostering Advanced Algebraic Thinking with Casio Technology Investigation 6.5—Earth’s Revolution Did You Know? Before the 1530s, many people believed the Earth was the center of the Universe and every celestial body revolved around it, including the sun! Copernicus, a Polish astronomer, presented a cosmological theory stating the Earth and other heavenly bodies actually revolved around the sun.
  • Page 101 Chapter 6: Conic Sections e. Use the calculator to verify the location of the foci determined in part a. Why might these answers vary slightly? f. What is the eccentricity of the ellipse? Discuss what that this means in general terms.
  • Page 102 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Earth’s Revolution a. Set up an axis system that can describe the Earth’s revolution about the Sun using the center of the major axis of the ellipse as the origin. What are the coordinates of the Earth at its perihelion and aphelion? The total distance along the major axis is 2A.
  • Page 103 Chapter 6: Conic Sections Sample Solution—Earth’s Revolution (continued) From this, we find that the sun is located at (-2.5 x 10 , 0). c. What are the coordinates of the Earth when it is an equal distance from both foci? The m inor vertices are equal distances from each focus.
  • Page 104 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Earth’s Revolution (continued) To find B, we used the calculator as shown below. We find that B is approximately 1.495 x 10 km, which we note is almost identical to A. d. Graph this ellipse and select an appropriate viewing window. Which view best...
  • Page 105 Chapter 6: Conic Sections Sample Solution—Earth’s Revolution (continued) However, the window is not yet squared, so we cannot tell how far removed from a circle this elliptical path really is. Here the ellipse appears elongated. Press (V-Window) , select (SQUARE) then (Y-BASE) .
  • Page 106 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Earth’s Revolution (continued) Though we note slight rounding errors, relative to the distances we are discussing, these are minor. What is noteworthy is how close to the center the two foci appear to be.
  • Page 107 Chapter 6: Conic Sections Investigation 6.6—Building Bridges Did You Know? The Golden Gate Bridge in San Francisco, California is a well known suspension bridge that was completed in 1937. It is approximately 9000 feet long and features 80,000 miles of wire in its two main cables. An average of 100,000 vehicles cross the Golden Gate Bridge each week.
  • Page 108 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Building Bridges Let’s investigate this from the Statistics menu: Press (SET UP), then, set the S S tat Wind to (Manual) . Scroll down to B B ackground, press (OPEN) to see which image files are available to load as a background.
  • Page 109 Chapter 6: Conic Sections Sample Solution—Building Bridges (continued) Continue this process until you have all 7 points. Then: Enter the x-values into L L ist 1 and the y-values into L L ist 2. If you have not entered the points in order from left to right, press , if necessary for more options and then (TOOL) .
  • Page 110 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Building Bridges (continued) Press (COPY) to copy the function into the GRAPH and TABLE menus (be sure not to overwrite any functions you wish to keep); even though no confirmation message appears, your function has been stored. Press (DRAW) to view your quadratic function overlaid on your scatterplot.
  • Page 111 Chapter 6: Conic Sections Sample Solution—Building Bridges (continued) To determine the x-values of the supports immediately to the left and right of the picture, we have chosen to subtract 0.88 from the left end, obtaining approximately 3.55, and adding 1.02 to the right end, obtaining 4.04. We want to substitute these values into our quadratic function to determine their corresponding y-values.
  • Page 112 Fostering Advanced Algebraic Thinking with Casio Technology Sample Solution—Building Bridges (continued) To find the height of the truck in the picture: Press (Sketch) , for more options, (PLOT) , and (Plot on) . Move the cursor to the top of the truck.