HW.1.1: Transformations Describe how each function is a transformation of the original function f ( x ) 1. f ( x 49). f( x+ 3) 3. f ( x ) + 5 4. f ( x) 5. f ( x ) + 3 11. Write a formula for f( x) 1. Write a formula for f( x) = x shifted up 1 unit and left units. = x shifted down 3 units and right 1 unit. 1 13. Write a formula for f( x) = shifted up units and left 4 units. x 14. Tables of values for f( x ), gx, ( ) and hx ( ) are given below. Write gx ( ) and hx ( ) as transformations of f ( x ). x - -1 0 1 f(x) - -1-3 1 x -1 0 1 3 g(x) - -1-3 1 x - -1 0 1 h(x) -1 0-3 15. Tables of values for f( x ), gx, ( ) and hx ( ) are given below. Write gx ( ) and hx ( ) as transformations of f ( x ). x - -1 0 1 f(x) -1-3 4 1 x -3 - -1 0 1 g(x) -1-3 4 1 x - -1 0 1 h(x) - -4 3 1 0 The graph of f ( x ) = x is shown. Sketch a graph of each transformation of f ( x ) x 17. g( x ) = + 1 wx = x 1 18. ( ) h x = 19. ( ) x x 0. g( x ) = + 1 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Sketch a graph of each function as a transformation of a toolkit function. f t = ( t+ 1) 3 3. ( ) h 4. ( x) = x 1 + 4 k 5. ( x) ( x ) 3 = 1 mt 6. ( ) = 3 + t + Write an equation for each function graphed below. 7. 8. 9. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
30. Starting with the graph of f ( x ) = 4 x write the equation of the graph that results from a. reflecting f ( x ) about the x-axis b. reflecting f ( x ) about the y-axis, shifting right 4 units, and up units Write an equation for each function graphed below. 31. 3. 33. Describe how each function is a transformation of the original function f ( x ). 37. f( x) 38. 4 f( x ) 39. f(5 x ) David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
1 40. f x 3 3 f x 41. ( ) Write a formula for the function that results when the given toolkit function is transformed as described. 51. f( x) = x reflected over the y axis and horizontally compressed by a factor of 1 4. 5. f( x) = x reflected over the x axis and horizontally stretched by a factor of. 1 53. f( x) = vertically compressed by a factor of 1, then shifted to the left units and down 3 units. x 3 Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. f x = 4 x+ 1 5 54. ( ) ( ) 55. h( x) = x 4 + 3 1 = x 56. ( ) 3 m x 57. p( x) 1 = x 3 3 58. a( x) = x+ 4 The function f ( x ) is graphed here. Write an equation for each graph below as a transformation of f ( x ). 59. 60. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
1. Consider the graph of ( ) HW..1: Complex Roots Visualization f x = x 4x+ 5 a. Graph f ( x ) in detail in the xy plane of the coordinate system provided on the next page without using a graphing calculator. b. What conclusion can you make about the roots of f ( x )? c. Show that i is a root for f ( x ). d. Show that f ( 3i) is a real number. e. Show that f ( 3 i) + is not a real number. f. Try a few more complex values and make a conjecture about values of a and b for f a+ bi is a real number. Explain how you arrived at your conjecture and which ( ) prove that it is true. g. Lastly, draw a graphical representation of what your above answer imply about the realvalued outputs of f with regard to the inclusion of a complex domain. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
f x = x + 8x+ 18. Consider the graph of ( ) a. Graph f ( x ) in detail in the xy plane of the coordinate system provided on the next page without using a graphing calculator. b. What conclusion can you make about the roots of f ( x )? c. Show that 4+ i is a root for f ( x ). d. Show that f ( 4 3i) is a real number. e. Show that f ( 3 i) + is not a real number. f. Try a few more complex values and make a conjecture about values of a and b for f a+ bi is a real number. Explain how you arrived at your conjecture and which ( ) prove that it is true. g. Lastly, draw a graphical representation of what your above answer imply about the realvalued outputs of f with regard to the inclusion of a complex domain. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
.3 Conic Sections: Ellipse Ellipse: (locus definition) set of all points (x, y) in the plane such that the sum of each of the distances from F 1 and F is d. Standard Form of an Ellipse: Horizontal Ellipse ( x h) ( y k) a + = 1 b center = ( hk, ) a = length of major axis b = length of minor axis Vertical Ellipse ( x h) ( y k) b + = 1 c = distance from center to focus c = a b c eccentricity e = ( 0< e < 1 the closer to 0 the more circular) a a David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Ex. Graph ( x+ 1 ) ( y ) + = 1 9 5 Center: ( 1, ) Endpoints of Major Axis: (-1, 7) & ( -1, -3) Endpoints of Minor Axis: (-4, -) & (, ) Foci: (-1, 6) & (-1, -) Eccentricity: 4/5 Ex. Graph x y x y + 4 + 4 + 33= 0 Homework: David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
1. Jamie and Jason want to exchange secrets (terrible secrets) from across a crowded whispering gallery. Recall that a whispering gallery is a room which, in cross section, is half of an ellipse. If the room is 40 feet high at the center and 100 feet wide at the floor, how far from the outer wall should each of them stand so that they will be positioned at the foci of the ellipse?. An elliptical arch is constructed which is 6 feet wide at the base and 9 feet tall in the middle. Find the height of the arch exactly 1 foot in from the base of the arch. 5. Can you write a circle in the standard form of an ellipse? When will an ellipse become a circle? David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
.3 Conic Sections: Hyperbola Hyperbola (locus definition) Set of all points x, y the difference of each distances from F 1 and F to x, y ( ) in the place such that the absolute value of ( ) is a constant distance, d. In the figure above: The distance from F 1 to x 1, y 1 F and The distance from F 1 to x, y F ( ) - the distance from to ( x 1, y 1 ) = d ( ) - the distance from to ( x, y ) = d Standard Form of a Hyperbola: Horizontal Hyperbola ( x h) ( y k) = 1 a b Vertical Hyperbola ( y k) ( x h) = 1 a b Horizontal Asymptotes ( ) center = h,k a = distance between vertices c = distance from center to focus c = a + b eccentricity e = c a ( e > 1 for a hyperbola) Vertical Asymptotes y k = ± b ( a x h) y k = ± a ( b x h) Show how d = a David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Ex. Graph ( x ) y 4 5 = 1 Center: (, 0) Vertices (4, 0) & (0, 0) ( ) Foci ± 9,0 Asymptotes: y = ± 5 ( x ) Ex. Graph 9y x 6x 10 = 0 Center: (-3, 0) Vertices 3, 1 3 & 3, 1 3 Foci 3,± 10 3 Asymptotes: y = ± 1 ( 3 x + 3) Hyperbolas can be used in so-called trilateration or positioning problems. The procedure outlined in the next example is the basis of the LOng Range Aid to Navigation (LORAN) system, (outdated now due to GPS) David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Ex. Jeff is stationed 10 miles due west of Carl in an otherwise empty forest in an attempt to locate an elusive Sasquatch. At the stroke of midnight, Jeff records a Sasquatch call 9 seconds earlier than Carl. Kai is also camping in the woods, he is 6 miles due north of Jeff and heard the Sasquatch call 18 seconds after Jeff did. If the speed of sound that night is 760 miles per hour, determine the location of the Sasquatch. Relationship between Jeff and Carl: 760 miles hour 1 hour 9 seconds=1.9 miles is the constant d, d = a 3600 seconds a = 0.95 c = 5 c = a + b, b = 4.0975 x 0.905 y 4.0975 = 1 Relationship between Jeff and Kai: miles 1 hour 760 18 seconds =3.8 miles is the constant d, d = a hour 3600 seconds a = 1.9 c = 3 b = 5.391 ( y 3) ( ) 3.61 x + 5 5.39 = 1 Using a graphing utility we find Sasquatch located at ( 0.969, 0.8113). Without a graphing utility each hyperbola would need to be written in the form Ax + Cy + Dx + Ey + F = 0 and use techniques for solving systems of non-linear equations, matrices. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Homework: David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
.3 Conic Sections Parabola Parabola (locus definition) Set of all points equidistant from a Focus to a Directrix. Standard Form of a Parabola: Vertical Parabola Horizontal Parabola ( x h) = 4p( y k) ( y k) = 4p( x h) vertex = (h, k) p = distance from vertex to focus David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Focus: ( hk, + p) Focus: ( h+ p, k) Directrix: y= k p Directrix: x= h p 4p = Latus Rectum = focal diameter of the parabola Ex. Graph ( x +1) = 8( y 3) Vertical Parabola 1, 3 Vertex: ( ) p = - Focus: ( 1,1) Directrix: y = 5 Ex. Consider the equation y + 4y+ 8x= 4. Put this equation into standard form and identify the vertex, focus, directrix, and graph. Vertex: ( 1, ) Focus: ( 1, ) Directrix: x = 3 Ex. A satellite dish is to be constructed in the shape of a paraboloid of revolution. If the receiver placed at the focus is located ft above the vertex of the dish, how deep will the dish be? David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Use (0, 0) as the vertex, (6, y) a point on the parabola, p =, and plug into the standard form of a vertical parabola. ( x 0) = 4 p( y 0) x = 4( )y ( 6) = 8y 36 8 = y 9 = y = 4.5 HW: David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
HW.4.: Using Matrices to Model Conics General Form for all Conics: conic. Ax + Bxy + Cy + Dx + Ey + F = 0, the Bxy term creates a rotated Write the Conic in General Form by using Matrices. On questions 1-5 the conics are not rotated so the Bxy term is not needed. Calculators are allowed. ( ) ( 1,4 ) 1.) Find the equation for a circle containing (6,1),,, and. Remember that on a circle A and C are equal, in order to solve with a Matrix set A and C equal to 1. ( ) (,1) ( 8,9) ( 9,) ( 10,5).) Find the equation of the circle that passes through 1,8,,,, and. You do not need to use all 5 points. ( ) ( 1,4 ) 3.) Find the equation for a parabola containing (6,1),,, and. This is a vertical parabola so the C value is 0, set the A value equal to 1 to solve. ( ) ( 1,4 ) 4.) Find the equation for a parabola containing (6,1),,, and. This is a horizontal parabola so the A value is 0, set the C value equal to 1 to solve. ( ) ( 1,4 ) ( 9,) 5.) Find the equation for an ellipse containing (6,1),,,, and. Set the A value equal to 1 and solve for C, D, E, and F. (Optional) Write the Conic is General Form by using Matrices. On questions 6-9 the conics are rotated so the Bxy term is needed. 7 6.) Find the equation of the conic that passes through,,,, and, 5 5 (,), 7 ( 7,5) Set the A value equal to 1. 19,7 14 7.) Find the equation of the conic that passes through,,,, and 5, 5 (,) 5,14 5 ( 7,5) 10,10. Set the A value equal to 1. ( ) 7 8.) Find the equation of the conic that passes through,,,, and 4,13 31 4 (,) 1, 7 13 4 4, 7 4 38 5,14 5. Set the A value equal to 1. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
In order to get your answer to match the answer below multiply everything by the Least Common Denominator, which makes the Conic in General Form (all coefficients are integers) Answers 1.) Circle: 7x + 7y 67x 65 + 08 = 0.) Circle: x + y 10x 10y + 5 = 0 3.) Parabola: 7x 61x 0y +134 = 0 4.) Parabola: 7y 6x 45y + 74 = 0 5.) Ellipse: 3x + 8y 33x 60y +14 = 0 6.) Rotated Parabola: x 4xy + y x y + 4 = 0 7.) Rotated Ellipse: 8.) Rotated Hyperbola: 17x 30xy +17y 4x 4y + 80 = 0 3x 10xy + 3y +16 = 0 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
John Mathews. http://mathfaculty.fullerton.edu/mathews/numerical/linear/cfo/cof.html HW.5.1: Linear Regression 1. The following is data for the first and second quiz scores for 8 students in a class. Plot the points, then sketch a line that fits the data. First Quiz 11 0 4 5 33 4 46 49 Second Quiz 10 16 3 8 30 39 40 49. Eight students were asked to estimate their score on a 10 point quiz. Their estimated and actual scores are given. Plot the points, then sketch a line that fits the data. Predicted 5 7 6 8 10 9 10 7 Actual 6 6 7 8 9 9 10 6 Based on each set of data given, calculate the regression line using your calculator or other technology tool, and determine the correlation coefficient. 3. x y 5 4 7 1 10 17 1 15 4 4. x y 8 3 15 41 6 53 31 7 56 103 5. x y 3 1.9 4. 5.74 6.6 7 0.78 8 17.6 9 16.5 10 18.54 11 15.76 1 13.68 13 14.1 14 14.0 15 11.94 16 1.76 17 11.8 18 9.1 6. x y 4 44.8 5 43.1 6 38.8 7 39 8 38 9 3.7 10 30.1 11 9.3 1 7 13 5.8 14 4.7 15 16 0.1 17 19.8 18 16.8 7. A regression was run to determine if there is a relationship between hours of TV watched per day (x) and number of situps a person can do (y). The results of the regression are given below. Use this to predict the number of situps a person who watches 11 hours of TV can do. y=ax+b a=-1.341 b=3.34 r =0.803 r=-0.896 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
8. A regression was run to determine if there is a relationship between the diameter of a tree (x, in inches) and the tree s age (y, in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches. y=ax+b a=6.301 b=-1.044 r =0.940 r=-0.970 Match each scatterplot shown below with one of the four specified correlations. 9. r = 0.95 10. r = -0.89 11. r = 0.6 1. r = -0.39 A B C D 13. The US census tracks the percentage of persons 5 years or older who are college graduates. That data for several years is given below. Determine if the trend appears linear. If so and the trend continues, in what year will the percentage exceed 35%? Year 199 0 199 199 4 199 6 199 8 00 0 00 00 4 00 6 00 8 Percent Graduate s 1.3 1.4. 3.6 4.4 5.6 6.7 7.7 8 9.4 14. The US import of wine (in hectoliters) for several years is given below. Determine if the trend appears linear. If so and the trend continues, in what year will imports exceed 1,000 hectoliters? Year 199 Imports 66 5 199 4 68 8 199 6 356 5 199 8 41 9 00 0 458 4 00 565 5 00 4 654 9 00 6 795 0 00 8 848 7 00 9 946 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
HW.5: Modeling Data Note to Instructors: The problems in this worksheet are already grouped by best-fit function family. It is suggested that you pick one from each category and rearrange so students utilize the R value and the scatterplot to determine the best regression function family. Linear Regression Problems 1. As Earth s population continues to grow, the solid waste generated by the population grows with it. Governments must plan for disposal and recycling of ever growing amounts of solid waste. Planners can use data from the past to predict future waste generation and plan for enough facilities for disposing of and recycling the waste. Given the following data on the waste generated in Florida from 1990-1994, how can we construct a function to predict the waste that was generated in the years 1995-1999? The scatter plot is shown in Figure 1.85. Year Tons of Solid Waste Generated (in thousands) 1990 19,358 1991 19,484 199 0,93 1993 1,499 1994 3,561 a) Make a scatterplot of the data, letting x represent the number of years since 1990. b) Use a graphing calculator to fit linear, quadratic, cubic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Graph the function of best fit with the scatterplot of the data. d) With each function found in part (b), predict the average tons of waste in 000 and 005, and determine which function gives the most realistic predictions.. The numbers of insured commercial banks y (in thousands) in the United States for the years 1987 to 1996 are shown in the table. (Source: Federal Deposit Insurance Corporation). Year 1987 1988 1989 1990 1991 199 1993 1994 1995 1996 y 13.70 13.1 1.71 1.34 11.9 11.46 10.96 10.45 9.94 9.53 Make a scatterplot of the data, letting x represent the number of years since 1987. a) Use a graphing calculator to fit linear, quadratic, cubic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. b) Graph the function of best fit with the scatterplot of the data. c) With each function found in part (b), predict the average number of insured commercial banks in 000 and 005, and determine which function gives the most realistic predictions. e) Plot the actual data and the model you selected on the same graph. How closely does the model represent the data? David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
3. U.S. Farms. As the number of farms has decreased in the United States, the average size of the remaining farms has grown larger, as shown in the table below. Year Average Acreage Per Farm 1910 139 190 149 1930 157 1940 175 1950 16 1959 303 1969 390 1978 449 1987 46 1997 487 a) Make a scatterplot of the data, letting x represent the number of years since 1900. b) Use a graphing calculator to fit linear, quadratic, cubic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Graph the function of best fit with the scatterplot of the data. d) With each function found in part (b), predict the average acreage in 000 and 010 and determine which function gives the most realistic predictions. 4. Sports The winning times (in minutes) in the women s 400-meter freestyle swimming event in the Olympics from 1936 to 1996 are given by the following ordered pairs. (1936,5.44) (197, 4.3) (1948,5.30) (1976, 4.16) (195,5.0) (1980, 4.15) (1956, 4.91) (1984, 4.1) (1960, 4.84) (1988, 4.06) (1964, 4.7) (199, 4.1) (1968, 4.53) (1996, 4.1) a) Make a scatterplot of the data, letting x represent the number of years since 197. b) Use a graphing calculator to fit linear, quadratic, cubic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Graph the function of best fit with the scatterplot of the data. d) Plot the actual data and the model you selected on the same graph. How closely does the model represent the data? Quadratic Regression Problems 1. The following data was obtained by throwing a rubber ball at a CBR. Time (sec) Height (m) 0.0000 1.03754 0.1080 1.4005 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
0.150 1.63806 0.35 1.7741 0.4300 1.8039 0.5375 1.715 0.6450 1.5094 0.755 1.1410 0.8600 0.83173 a) Use the data above to make a scatterplot, letting x represent the number of seconds elapsed. b) Next, use a graphing calculator to find the model that best expresses the height and vertical velocity of the rubber ball. We can also use this model to predict the maximum height of the ball and its vertical velocity when it hits the face of the CBR. c) Fit linear, quadratic, cubic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. d) Graph the function of best fit with the scatterplot of the data. e) Determine the maximum height of the ball (in meters). f) With the model you selected in part (b), predict when the height of the ball is at least 1.5 meters.. Stopping Distance A state highway patrol safety division collected the data on stopping distances in Table.16. a) Draw a scatter plot of the data. b) Fit linear, quadratic, cubic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the regression model to predict the stopping distance for a vehicle traveling at 5 mph. e) Use the regression model to predict the speed of a car if the stopping distance is 300 ft. Table.16 Highway Safety Division Speed (mph) Stopping Distance (ft) 10 15.1 0 39.9 30 75. 40 10.5 50 175.9 3. Home Schooling Growth The estimated number of U.S. children that were home-schooled in the years from 199 to 1997 were: Table 1.13 Home Schooling Year Number 199 703,000 1993 808,000 1994 99,000 1995 1,060,000 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
1996 1,0,000 1997 1,347,000 (a) Produce a scatter plot of the number of children home-schooled in thousands (y) as a function of years since 1990 (x). (b) Find the linear regression equation. (Round the coefficients to the nearest 0.01.) (c) Does the value of r suggest that the linear model is appropriate? (d) Find the quadratic regression equation. (Round the coefficients to the nearest 0.01.) (e) Does the value of R suggest that a quadratic model is appropriate? (f) Use both curves to predict the number of U.S. children that are home-schooled in the year 005. How different are the estimates? (g) Writing to Learn Use the results of this exploration to explain why it is risky to use regression equations to predict y-values for x values that are not very close to the data points, even when the curves fit the data points very well. 4. Leisure Time The following table shows the median number of hours of leisure time that Americans had each week in various years. Year Median Number of Leisure Hours Per Week 1973, 0 6. 1980, 7 19. 1987, 14 16.6 1993, 0 18.8 1997, 4 19.5 Source: Louis Harris and Associates (a) Make a scatterplot of the data, letting x represent the number of years since 1973, and determine which model best fits the data. (b) Use a graphing calculator to fit the type of function determined in part (a) to the data. (c) Graph the equation with the scatterplot. Then, use the function found in part (b) to estimate the number of leisure hours per week in 1978,1990, and 005. 5. On-line Travel Revenue With the explosion of increased Internet use, more and more travelers are booking their travel reservations on-line. The following table lists the total on-line revenue for recent years. Most of the revenue is from airline tickets. Year On-Line Travel Revenue (In Millions) 1996 $ 76 1997 87 1998 1900 1999 300 000 4700 001 6500 00 8900 Source: Travel and Interactive Technology 1999 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
(a) Create a scatterplot of the data. Let x= the number of years since 1996. (b) Use a graphing calculator to fit the data with linear, quadratic, and exponential functions. Determine which function has the best fit. (c) Graph all three functions found in part (b) with the scatterplot in part (a). (d) Use the functions found in part (b) to estimate the on-line travel revenue in 010. Which function provides the most realistic prediction? Quartic Regression Problems 1. Consumer Debt Nonmortgage consumer debt is mounting in the United States, as shown in the table below. Year Non-mortgage Debt (In Billions) 1989 $ 76 1990 789 1991 783 199 775 1993 804 1994 90 1995 1038 1996 1161 1997 116 1998 166 f) Draw a scatter plot of the data. g) Fit linear, exponential, power, cubic, and quartic functions to the data. By comparing the values of R, determine the function that best fits the data. h) Superimpose the regression curve on the scatter plot. i) Use the regression model to predict when consumer debt will reach 1400 billion dollars.. Declining Number of Farms in the United States Today U.S. farm acreage is about the same as it was in the early part of the twentieth century, but the number of farms has shrunk. Year Number of Farms (in millions) 1910 6.4 190 6.5 1930 6.3 1940 6.1 1950 5.4 1959 3.7 1969.7 1978.3 1987.1 1997 1.9 Looking at the table above, we note that the data could be modeled with a cubic or a quartic function. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
(a) Model the data with both cubic and quartic functions. Let the first coordinate of each data point be the number of years after 1900. That is, enter the data as (10, 6.4), (0, 6.5), and so on. Then using R, the coefficient of determination, decide which functions is the better fit. The R -value gives an indication of how well the function fits the data. The closer R is to 1, the better the fit. (b) Graph the function with the scatterplot of the data. (c) Use the answer to part (a) to estimate the number of farms in 1900, 1975, and 003. Exponential Regression Problems 1. In the years before the Civil War, the population of the United States grew rapidly, as shown in the following table from the U.S. Bureau of the Census. Year Population in Millions 1790 3.93 1800 5.31 1810 7.4 180 9.64 1830 1.86 1840 17.07 1850 3.19 1860 31.44 a) Draw a scatter plot of the data. b) Fit linear, quadratic, exponential, power, logarithmic, and logistic functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the regression model to predict the population in 1870. e) Use the regression model to predict the population in 1930. Explain why/why not you feel this prediction has validity. (Hint: you may want to complete this problem after you finish the problem dealing with Census records after the Civil War.). Projected Number of Alzheimer s Patients: German psychiatrist Alois Alzheimer first described the disease, later called Alzheimer s disease, in 1906. Since life expectancy has significantly increased in the last century, the number of Alzheimer s patients has increased dramatically. The number of patients in the United States reached 4 million in 000. The following table lists projected data regarding the number of Alzheimer s patients in years beyond 000. a) Draw a scatter plot of the data. Year, x Projected Number of Alzheimer s Patients in the United States (In millions) 000 4.0 010 5.8 00 6.8 030 8.7 040 11.8 050 14.3 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
b) Fit linear, exponential, power, logistic and logarithmic functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the regression model to estimate the number of Alzheimer s patients in 005, 05, and 100. 3. Number of physicians: The following table contains data regarding the number of physicians in the United States in selected years. Year Total Number of Physicians 1950 19,997 1955 41.711 1960 60.484 1965 9,088 1970 334,08 1975 393,74 1980 467,679 1985 55,716 1990 615,41 1994 684,414 1995 70,35 1996 737,764 a) Draw a scatter plot of the data. b) Fit linear, quadratic, cubic, exponential, quartic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the regression model to predict the population in 1975. e) Use the regression model to estimate the number of physicians in 000 and 05. 3. Credit Card Volume: The total credit card volume for Visa, MasterCard, American Express, and Discover has increased dramatically in recent years, as shown in the table below. (Source, CardWeb Inc. s CardData) Year, x Credit Card Volume, y (In Billions) 1988 61.0 1989 96.3 1990 338.4 1991 361.0 199 403.1 1993 476.7 1994 584.8 1995 701. 1996 798.3 1997 885. a) Draw a scatter plot of the data. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
b) Fit linear, quadratic, cubic, exponential, quartic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the regression model to predict the credit card volume in 003 and in 010. Logarithmic Regression Problems 1. Forgetting In an art class, students were tested at the end of the course on a final exam. Then they were retested with an equivalent test at subsequent time intervals. Their scores after time t, in months, are given in the table. Time, t (in Score, y months) 1 84.9% 84.6% 3 84.4% 4 84.% 5 84.1% 6 83.9% a) Draw a scatter plot of the data. b) Fit linear, quadratic, logarithmic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the regression model to predict test scores after 8, 10, 4, and 36 months. e) After how long will the test scores fall below 8%? Jamie, a meteorologist, is interested in finding a function that explains the relation between the height of a weather balloon (in kilometers) and the atmospheric pressure (measured in millimeters of mercury) on the balloon. She collects the data shown in Table 10. Table 10 Atmospheric Pressure, p Height,h 760 0 740 0.184 75 0.38 700 0.565 650 1.079 630 1.91 600 1.634 580 1.86 550.35 a) Using a graphing utility, draw a scatter diagram of the data with atmospheric pressure as the independent variable. b) Fit linear, quadratic, logarithmic, and power functions to the data. By comparing the values of R, determine the function that best fits the data. c) Superimpose the regression curve on the scatter plot. d) Use the function in part (b) to predict the height of the weather balloon if the atmospheric pressure is 560 millimeters of mercury. 3. Economics and Marketing The following data represent the price and quantity supplied in 005 for IBM personal computers. Price ($/Computer) Quantity Supplied 300 180 000 173 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
1700 160 1500 150 1300 137 100 130 1000 113 (a) Using a graphing utility, draw a scatter diagram of the data with price as the dependent variable. (b) Using a graphing utility, try a variety of function families. Compare the values R to find the function that best fits the data. (c) Using a graphing utility, draw the function found in part (b) on the scatter diagram. (d) Use the function found in part (b) to predict the number of IBM personal computers that will be supplied if the price is $1650. Power Regression Problems 1. Use the data in the table below to obtain a model for speed p versus distance traveled d. Consider linear, quadratic, exponential, power, and quartic models. Then use the model you selected as the best fit to predict the speed of the ball at impact, given that impact occurs when d 1.80 m. Table.1 Rubber Ball Data from CBR Experiment Distance (m) Speed (m/s) 0.00000 0.00000 0.0498 0.837 0.16119 1.71163 0.35148.45860 0.59394 3.0509 0.89187 3.7400 1.5557 4.49558. The length of time that a planet takes to make one complete rotation around the sun is its year. The table shows the length (in earth years) of each planet s year and the distance of that planet from the sun (in millions of miles). Find a model for this data in which x is the length of the year and y the distance from the sum. Planet Year Distance Mercury.4 36.0 Venus.6 67. Earth 1 9.9 Mars 1.88 141.6 Jupiter 11.86 483.6 Saturn 9.46 886.7 Uranus 84.01 1783.0 Neptune 164.79 794.0 David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
Pluto 47.69 3674.5 3. Cholesterol Level and the Risk of Heart Attack. The data in the following table show the relationship of cholesterol level in men to the risk of a heart attack. Cholesterol Level, x Men, Per 10,000, Who Suffer A Heart Attack, y 100 30 00 65 50 100 75 130 300 180 (a) Use a graphing calculator to fit a model function to the data. Consider linear, exponential, power, and cubic functions. (b) Graph the function with the scatterplot of the data. (c) Use the answer to part (a) to estimate the heart attack rate for men with cholesterol levels of 150, 350, and 400. Logistic Regression Problems 1. After the Civil War, the U.S. population increased, as shown below. Year Population in Millions 1870 38.56 1880 50.19 1890 6.98 1900 76.1 1910 9.3 190 106.0 1930 13.0 1940 13.16 1950 151.33 1960 179.3 1970 0.30 1980 6.54 1990 48.7 000 81.4 a) Draw a scatter plot of the data. b) Fit linear, quadratic, exponential, power, logarithmic, and logistic functions to the data. By comparing the values of R, determine the function that best fits the data. c) Use the regression model to predict the population in 1975 and in 010. Explain why/why not you feel this prediction has validity.. Effect of Advertising A company introduces a new software product on a trial run in a city. They advertised the product on television and found the following data relating the percent P of people who bought after x ads were run. Number of Ads, x % Who Bought, P 0 0. David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.
10 0.7 0.7 30 9. 40 7 50 57.6 60 83.3 70 94.8 80 98.5 90 99.6 Draw a scatter plot of the data. Then, fit linear, exponential, power, logistic and logarithmic functions to the data. By comparing the values of R, determine the function that best fits the data. Then use the regression model to predict the percent P of people who will buy the software after 100 ads are run. * Relate what you have discovered in this exercise to what you have observed in television ads. What could the company do to change this pattern? David Lippman and Melonie Rasmussen 015. Precalculus: An Investigation of Functions Ed. 1.5.