C H A P T ER Polynomial Functions and Data Modeling Daniel Gabriel Fahrenheit (1686 1736) was a Prussian physicist and engineer who determined a temperature scale named after him. Swedish astronomer Anders Celsius (1701 17) developed a similar temperature scale. Water freezes at 32 on the Fahrenheit scale and 0 on the Celsius scale. You will use compositions of functions to convert temperatures between the Celsius and Fahrenheit scales..1 Cold and Colder! Function Composition p. 157.2 Pieces of Pieces Piecewise Polynomial Functions p. 169.3 Model Fitting! Modeling with Polynomial Functions p. 175. More Model Fitting! Modeling with Polynomial Functions p. 183 Chapter l Polynomial Functions and Data Modeling 155
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.1 Cold and Colder! Function Composition Objectives In this lesson you will: l Evaluate functions and compositions of functions. l Determine the inverse of a function using inverses. l Write functions as compositions of functions. l Describe the relationship between the graph of a function, and the graph of its inverse with respect to the line y x. Key Term l composition of functions Problem 1 Temperature Scales Two common temperature scales are the Fahrenheit scale and the Celsius scale. Both of these scales are based on the freezing point of water, which is 32 F and 0 C. The following formulas can be used to convert between the two scales: l F 9 C 32 5 l C 5 (F 32) 9 where F is the temperature in degrees Fahrenheit and C is the temperature in degrees Celsius. 1. Calculate each unit conversion. a. Convert 50 C to degrees Fahrenheit. b. Convert 75 C to degrees Fahrenheit. c. Convert 131 F to degrees Celsius. d. Convert 80 F to degrees Celsius. Lesson.1 l Function Composition 157
Two lesser known temperature scales are the Kelvin scale and the Rankine scale. Generally, scientists use both of these temperature scales. Both the Kelvin and Rankine scales are based on absolute zero, the coldest temperature that is theoretically possible, approximately 273.15 C. The Kelvin and Rankine scales do not use degrees. One kelvin unit, written as K, is equal to 1 C. One Rankine unit, written as R, is equal to 1 F. Therefore, absolute zero is equal to 0 K and 0 R. 2. Convert absolute zero to degrees Fahrenheit. 3. Complete the table with the appropriate conversion formulas. To From Fahrenheit Celsius Kelvin Rankine Fahrenheit C 5_ (F 32) 9 Celsius F 9_ C 32 5 Kelvin Rankine. Calculate each unit conversion. a. Convert 100 F to Rankines and kelvins. b. Convert 20 C to Rankines and kelvins. c. Convert 100 R to degrees Celsius and degrees Fahrenheit. 158 Chapter l Polynomial Functions and Data Modeling
Problem 2 Composition of Functions Composition of functions is substituting one function in for the variable of another function. The composition of functions f and g is written as g f, or g (f (x)). 1. Write each temperature conversion as a function. a. Let f(x) represent the conversion from degrees Fahrenheit to degrees Celsius. b. Let g(x) represent the conversion from kelvins to degrees Celsius c. Let h(x) represent the conversion from Rankine to degrees Celsius. 2. Evaluate each function. a. f (10) b. g(10) c. f g(20) d. g f(20) Lesson.1 l Function Composition 159
e. f g h(0) 3. Write each temperature conversion as a function. a. Let f 1 (x) represent the conversion from degrees Celsius to degrees Fahrenheit. b. Let g 1 (x) represent the conversion from kelvins to degrees Fahrenheit. c. Let h 1 (x) represent the conversion from Rankines to degrees Fahrenheit.. Evaluate each composition using functions from Questions 1 and 3. Take Note Two functions f and g are inverses if f consists of a set of ordered pairs (x, y) and g consists of all corresponding ordered pairs ( y, x). If f and g are inverse functions then f (g(x)) g (f(x)) x for all values of x. a. f f 1 (x) b. f 1 f(x) 160 Chapter l Polynomial Functions and Data Modeling
5. What is the relationship between f and f 1? How do you know? 6. What other pairs of functions from the table have this relationship? 7. Let f(x) 3x, g(x) 2 x 2, and h(x) x3 1. Evaluate each composition. a. f g(x) b. g(f(x)) c. f h(x) d. f g h(x) Lesson.1 l Function Composition 161
A function can be defined as the composition of two or more functions. For example, the function f(x) can be defined as the composition of the functions g(x) and h(x) as follows. f(x) 2x 2 1 g(x) x 2 h(x) 2x 1 f(x) h g(x) 2(x 2 ) 1 2x 2 1 8. Define each function as the composition of two or more functions. a. f(x) x 2 6x 9 b. f(x) x 16x 2 162 Chapter l Polynomial Functions and Data Modeling
Problem 3 Composition of Functions, Inverses, and Symmetry To determine the inverse of a function, write a composition of the function and its inverse that equals x. Then, solve for the inverse function. For example, determine the inverse of the function f(x) 2x 5 as follows. f f 1 (x) x f (f 1 (x)) x 2f 1 (x) 5 x 2f 1 (x) x 5 f 1 (x) x 5 2 1. Graph and label f (x) x. 2. Determine f 1 (x). 3. If the ordered pair (a, b) satisfies f(x), then (b, a) also satisfies f(x). Why? Lesson.1 l Function Composition 163
. Identify one point that lies on the graph of f(x). What corresponding point lies on the graph of the inverse of f(x)? 5. Determine the equation of the line that passes through these two points. 6. Graph the line from Question 5. How is the graph of this line related to the graph of the line y x? Explain. 7. Calculate the point of intersection of the line from Question 5 and the line y x. 16 Chapter l Polynomial Functions and Data Modeling
8. Calculate the midpoint of the line segment with endpoints at the points from Question. 9. What do you notice about the points you calculated in Questions 7 and 8? 10. Graph g(x) x 2. y = x 11. Determine g 1 (x). Then, graph and label the inverse function. Lesson.1 l Function Composition 165
12. Identify one point that lies on the graph of g(x). What corresponding point lies on the graph of the inverse of g(x)? 13. Determine the equation of the line that passes through these two points. 1. Graph the line from Question 13. How is the graph of this line related to the graph of the line y x? Explain. 15. Calculate the point of intersection of the line from Question 13 and the line y x. 166 Chapter l Polynomial Functions and Data Modeling
16. Calculate the midpoint of the line segment with endpoints at the points from Question 12. 17. What do you notice about the points you calculated in Questions 15 and 16? 18. Describe the relationship between a point on the graph of a function and the corresponding point on the graph of its inverse with respect to the line y x. 19. Describe the relationship between the graph of a function and the graph of its inverse with respect to the line y x. Be prepared to share your methods and solutions. Lesson.1 l Function Composition 167
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.2 Pieces of Pieces Piecewise Polynomial Functions Objectives In this lesson you will: l Analyze graphs of piecewise polynomial functions. l Graph piecewise polynomial functions. l Define piecewise polynomial functions. Key Term l piecewise functions Problem 1 On You Mark, Get Set... The following graph displays a runner s speed, in kilometers per hour, during a ten kilometer (10K) race. y 18 16 1 Speed (km/h) 12 10 8 6 2 1 2 3 5 6 7 8 9 Distance Run (km) 1. Write a paragraph using the graph shown to describe the runner s speed during the 10K race. x Lesson.2 l Piecewise Polynomial Functions 169
2. At what point in the race was she running the fastest? What was her speed at that point? 3. What was her speed throughout most of the race?. Acceleration is the change in speed over time. Acceleration is measured in kilometers per second per second, or kilometers per second squared (km/s 2 ). During what interval did she have the greatest acceleration? 5. During what interval did she have the least acceleration? 6. What was her acceleration between the 1st and 9th kilometer? 7. What was her average acceleration for the first half kilometer of the race? What was her average acceleration for the last half kilometer? 170 Chapter l Polynomial Functions and Data Modeling
Problem 2 Breaking it Down The following graph displays the runner s speed during the first kilometer. y 18 16 1 Speed (km/h) 12 10 8 6 2 0.2 0. 0.6 0.8 1.0 1.2 1. 1.6 1.8 Distance Run (km) x 1. Describe the runner s speed during the first 1.1 km of the race. 2. Describe her speed during the first 0.5 kilometer of the race. 3. Describe her speed between 0.5 kilometer and 1 kilometer into the race.. Describe her speed after the first 1 kilometer into the race. Lesson.2 l Piecewise Polynomial Functions 171
5. What type of function could model her speed during each part of the race? a. the first 0.5 kilometer b. between 0.5 kilometer and 1 kilometer c. after 1 kilometer The following graph displays the runner s speed during the last kilometer. y 18 16 1 Speed (km/h) 12 10 8 6 2 x 8.2 8. 8.6 8.8 9.0 9.2 9. 9.6 9.8 10.0 Distance Run (km) 6. Describe her speed during the last kilometer of the race. 7. What type of function could model her speed during the last kilometer of the race? 172 Chapter l Polynomial Functions and Data Modeling
Problem 3 Piecewise Polynomials 1. Graph each piecewise function. 3x 1 x 0 a. f(x) x 3 x 0 Take Note A piecewise function is a function that has different rules for different parts of its domain. b. Graph g(x) x3 1 x 2 1 3 x 0 0 x 2 2 x Lesson.2 l Piecewise Polynomial Functions 173
c. h(x) x2 8x 12 x 2 x 3 6x 2 11x 5 x 2 2 x 1 1 x 2. The graph of g(x) is shown. Define g(x). y 8 6 2 8 6 2 2 6 2 6 8 x 8 Be prepared to share your methods and solutions. 17 Chapter l Polynomial Functions and Data Modeling
.3 Model Fitting! Modeling with Polynomial Functions Objectives In this lesson you will: l Determine linear functions determined by two points. l Determine quadratic functions determined by three points. l Use polynomial functions to model data. Problem 1 Polynomial Functions and Previously, you determined the equation of the line that passes through Distinct Points Previously, you determined the equation of the line that passes through two distinct points. You also determined the equation of a parabola that passes through three distinct nonlinear points. 1. Determine the equation of the line that passes through each pair of points. a. (, 3), ( 5, 8) b. (2, ), (5, 1) Lesson.3 l Modeling with Polynomial Functions 175
2. Determine the equation of a quadratic function y ax 2 bx c that passes through the points (2, 6), ( 1, 1), and ( 2, 5). a. Write a system of three equations in three unknowns by substituting the values from the ordered pairs in the general equation. b. Solve the system of equations using matrices. c. Graph the three points and the quadratic function. 176 Chapter l Polynomial Functions and Data Modeling
3. Determine the equation of the cubic function that passes through the points: (2, 5), ( 1, 1), ( 2, 5) and (3, 1) using technology. Graph the four points and the cubic function. This process can be used to determine the equation of a degree n polynomial that passes through n 1 distinct points. Lesson.3 l Modeling with Polynomial Functions 177
Problem 2 Modeling Data with Polynomial Functions For large data sets, it is not desirable to determine a function that passes through all the data points. For example, you would need a 23rd degree polynomial to perfectly model the following data set with 2 data points. Instead, you can determine a function that best fits a data set, even if it doesn t pass through every data point. Time (days) Average Salinity (parts per thousand) 31.5 Average Salinity 1 28.75 2 28.35 3 28.55 29.825 5 29.2 6 28.75 7 27.95 8 28.025 9 28.75 10 29.925 11 29.825 12 29.8 13 29. 1 29.75 15 29.375 16 29.25 17 29.35 18 29.225 19 29.2 20 29.275 21 29.225 22 29.15 23 29.1 2 29 Salinity (parts per thousand) 30.5 29.5 28.5 27.5 26.5 0 5 10 15 20 25 Time (days) 178 Chapter l Polynomial Functions and Data Modeling
For some data sets, it may be undesirable to use a single function to model the data. Piecewise functions can be useful to better model data. Consider the first 10 data points of the salinity data set. Time (days) Average Salinity (parts per thousand) 1 28.75 2 28.35 3 28.55 29.825 5 29.2 6 28.75 7 27.95 8 28.025 9 28.75 10 29.925 1. Enter the first 10 data points into a graphing calculator. Create a scatter plot using STAT PLOT function. 2. Determine the quadratic regression equation using the STAT CALC function. 3. Determine the cubic regression equation using the STAT CALC function.. Determine the quartic regression equation using the STAT CALC function. Lesson.3 l Modeling with Polynomial Functions 179
5. Graph each regression equation on the grid from Question 1. Which function is the best fit for the first 10 data points? Explain. Consider the last 1 data points of the salinity data set. Time (days) Average Salinity (parts per thousand) 11 29.825 12 29.8 13 29. 1 29.75 15 29.375 16 29.25 17 29.35 18 29.225 19 29.2 20 29.275 21 29.225 22 29.15 23 29.1 2 29 6. Enter the last 1 data points into L 3 and L of a graphing calculator. Create a scatter plot using STAT PLOT function. Be sure to turn off the other three functions. 180 Chapter l Polynomial Functions and Data Modeling
7. What type of function is the best fit for the last 1 data points? Explain. 8. Determine the regression equation using the STAT CALC function. Graph the regression equation on the grid from Question 6. 9. Define a piecewise function to model the complete data set. 10. Graph all the points and the piecewise function. Be prepared to share your methods and solutions. Lesson.3 l Modeling with Polynomial Functions 181
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. More Model Fitting! Modeling with Polynomial Functions Objective In this lesson you will: l Use polynomial functions to model data. Problem 1 Mean Price of Houses The following table displays the median and mean price of a single family home over a fifteen month period. Time (months) Median Price of a House in U.S. ($) Mean Price of House in U.S. ($) 1 29,100 316,800 2 227,700 28,00 3 232,00 28,600 25,300 301,200 5 259,300 301,600 6 26,00 31,300 7 25,300 305,200 8 239,300 299,00 9 237,300 301,900 10 229,000 295,500 11 225,200 287,100 12 213,500 277,000 13 221,600 268,100 1 229,600 263,100 15 238,600 25,200 Lesson. l Modeling with Polynomial Functions 183
1. Create a scatter plot of mean price over time. 2. What type of polynomial function best fits the data? Explain. 3. Determine the regression equation using a graphing calculator. Graph the regression equation on the grid from Question 1.. Over what interval does the regression model fit the data well? Over what interval does the regression model not fit the data well? 5. What type of polynomial function best fits the data representing the first seven months? Explain. 6. Determine a regression equation for the first seven months using a graphing calculator. Graph the regression equation on the grid from Question 1. 18 Chapter l Polynomial Functions and Data Modeling
7. Define a piecewise function to model the complete data set. 8. Graph the piecewise function. Problem 2 Median Price of Houses 1. Create a scatter plot of median price over time. 2. What type of polynomial function best fits the data? Explain. Lesson. l Modeling with Polynomial Functions 185
3. Determine the regression equation using a graphing calculator. Graph the regression equation on the grid from Question 1.. How well does the regression model fit the data? Explain. 5. Would a piecewise function better model the data? Explain. 6. Determine a piecewise function consisting of two or more polynomial functions that best fits the data. 7. Graph the piecewise function. Be prepared to share your methods and solutions. 186 Chapter l Polynomial Functions and Data Modeling