Properties of Elementary Functions

Transcription

1 2 Properties of Elementar Functions If a mother mouse is twice as long as her offspring, then the mother s weight is about eight times the bab s weight. But the mother mouse s skin area is onl about four times the bab s skin area. So the bab mouse must eat more than the mother mouse in proportion to its bod weight to make up for the heat loss through its skin. In this chapter ou ll learn how to use functions to model and eplain situations like this. CHAPTER OBJECTIVES Discover patterns in the graphs of linear, quadratic, power, and eponential functions. Given the graph of a function, know whether the function is eponential, power, quadratic, or linear and find the particular equation algebraicall. Given a set of regularl spaced -values and the corresponding -values, identif which tpe of function the fit (linear, quadratic, power, or eponential). Find other function values without necessaril finding the particular equation. Learn the properties of base- logarithms. Use logarithms with base or other bases to solve eponential or logarithmic equations. Show that logarithmic functions have the multipl add propert, and find particular equations algebraicall. Fit a logistic function to data for restrained growth

2 Chapter 2 Properties of Elementar Functions Overview In Chapter 2, students learn to tell which kind of function might fit a given set of data b recognizing first the geometric pattern of the graph and then the numerical pattern revealed b regularl spaced points. The numerical patterns include the add-multipl propert for eponential functions and the multiplmultipl propert for power functions. As the stud logarithms, students learn that logarithmic functions have the multipl-add propert. Students learn a verbal wa to remember the definition of logarithm, namel that a logarithm is an eponent. Natural logarithms and common logarithms are presented. The chapter concludes with the modeling of restrained population growth with the logistic function. Using This Chapter At this point ou ma prefer to postpone Chapters 2 4 and skip to the stud of periodic functions in Chapters 9. In this chapter students learn about graphical and numerical patterns associated with linear, eponential, and power functions. The logarithmic and logistic functions, important topics for calculus, are also studied. Chapter 2 sets the stage for Chapter 3, where functions are eamined using regression to find models for data. Section 2-3, Identifing Functions from Numerical Patterns, can be omitted without creating challenges later on for ou and our students. If students will not continue on to calculus, then Section 2-7, Logistic Functions for Restrained Growth, ma also be omitted. Teaching Resources Eplorations Eploration 2-2: Graphical Patterns in Functions Eploration 2-3: Patterns for Quadratic Functions Eploration 2-3a: Numerical Patterns in Function Values Eploration 2-3b: Equations from Given Values Practice Eploration 2-4: Introduction to Logarithms Eploration 2-4a: Introduction to Logarithmic Functions Eploration 2-7: The Logistic Function for Population Growth Eploration 2-8a: Rehearsal for Chapter 2 Test Blackline Masters Sections 2-7 to 2-8 Supplementar Problems Sections 2-2, 2-3, and 2- to 2-7 Assessment Resources Test 4, Sections 2-1 to 2-3, Forms A and B Test, Sections 2-4 to 2-7, Forms A and B Test 6, Chapter 2, Forms A and B Technolog Resources Sketchpad Presentation Sketches Logistic Present.gsp Activities Fathom: Moore s Law Fathom: Population Growth CAS Activit 2-4a: Dilations of Eponential Functions CAS Activit 2-a: Dilations of Logarithmic Functions 63A Chapter 2 Interleaf: Properties of Elementar Functions

3 Standard Schedule Pacing Guide Da Section Suggested Assignment Shapes of Function Graphs RA, Q1 Q, 1 2 odd Identifing Functions from Graphical Patterns 2 even, even Optional 4 Identifing Functions from Numerical RA, Q1 Q, 1 23 odd 2-3 Patterns 2 27, 29 32, Properties of Logarithms RA, Q1 Q, 1 47 odd 7 2- Logarithms: Equations and Other Bases RA, Q1 Q, 1, 2, 3 49 odd Logarithmic Functions RA, Q1 Q, 1 13 odd, Logistic Functions for Restrained Growth RA, Q1 Q, 1, 3, 7 R0 R7, T1 T Chapter Review and Test 11 Problem Set 3-1 Block Schedule Pacing Guide Da Section Suggested Assignment Identifing Functions from Graphical Patterns RA, Q1 Q, Identifing Functions from Numerical Patterns RA, Q1 Q, 1 24 multiples of 3, 2, 30, Properties of Logarithms RA, Q1 Q, 1-47 odd 4 2- Logarithms: Equations and Other Bases RA, Q1 Q, 1 7 odd, 1 37 odd, Logarithmic Functions RA, Q1 Q, 1 9 odd 2-7 Logistic Functions for Restrained Growth RA, 1, Logistic Functions for Restrained Growth Q1 Q,, Chapter Review and Test R0 R7, T1 T Chapter Review and Test 3-1 Introduction to Regression for Linear Data 1 6 Chapter 2 Interleaf 63B

4 Section 2-1 Class Time 1 2 da PLANNING Homework Assignment Problems 1 4 TEACHING Important Terms and Concepts Concavit (concave up, concave down) Eponential function Power function Quadratic function Linear function Section Notes In this section, students investigate the shapes and features of the graphs of linear, quadratic, power, and eponential functions. You can assign this section for homework the da of the Chapter 1 test or as a group activit to be completed in class. No classroom discussion is needed before students begin the activit. Differentiating Instruction Pass out the list of Chapter 2 vocabular, available at for ELL students to look up and translate in their bilingual dictionaries. ELL students should do the Eplorator Problem Set in pairs. The ma need more time than other students. Let them write in their primar language, English, or a combination of the two. PROBLEM NOTES Problems 1 and 2 present an eponential function and a power function, respectivel. Students sometimes confuse these two tpes of functions because Mathematical Overview ALGEBRAICALLY NUMERICALLY GRAPHICALLY VERBALLY 64 Chapter 2: In this chapter ou will etend what ou have alread learned about some of the more familiar functions in algebra, as well as some ou ma not et have encountered. These functions are You will stud these functions in four was. You can define each of these functions algebraicall; for eample, the logarithmic function is defined lo g b if and onl if b You can find interesting numerical relationships between the values of variables and propert: As a result of adding a constant to, the corresponding -value is multiplied b a constant. Population both involve an eponent. In Problem 1, emphasize that in an eponential function the variable is the eponent, whereas in a power function the variable is the base. In addition, ou ma want to discuss these questions with our class. a. The f ()-intercept is f (0), the value of f () when 0. What is the graphical significance of the f ()-intercept? Wh does the f ()-intercept equal 0.2, not 0, for this function? Logistic function Time Eponential functions can describe unrestrained population growth, such as that of rabbits if the have no natural enemies. Logistic functions start out like eponential functions but then level off. Logistic functions can model restrained population growth where there is a maimum sustainable population in a certain region. b. The word concave means hollowed out. The words cave and cavit have the same origin. Wh do ou suppose the graph of f is said to be concave up? c. B calculation, f (21) 0.1. Wh is f (21) not meaningful in this real-world situation? d. Wh do ou suppose the function in this problem is called an eponential function? 64 Chapter 2: Properties of Elementar Functions

5 f () 2-1 Objective Figure 2-1a Eplorator Problem Set 2-1 Shapes of Function Graphs In this chapter ou ll learn was to find a function to fit a real-world situation when the tpe of function has not been given. You will start b refreshing our memor about graphs of functions ou studied in Chapter Eponential Function Problem: In the eponential function f () 0.2 2, f () could be the number of thousands of bacteria in a culture as a function of time,, in hours. Find f () for and graph the function as in Figure 2-1b. The number of bacteria is increasing as time goes on. How does the concavit of the graph tell ou that the rate of growth is also increasing? 2. Power Function Problem: In the power function g () 0.1 3, g () could be the weight in pounds of a snake that is for each foot from 0 through 6, and graph function g. Because the graphs of f 1 and g concave up, what graphical evidence could ou use to distinguish between the two tpes of functions? Is the following statement true or false? The snake s weight increases b the same amount for each foot it increases in length. Give evidence to support our answer. Discover patterns in the graphs of linear, quadratic, power, and eponential functions. Figure 2-1a shows the plot of points that are values of the eponential function f () You can make such a plot b storing the -values in one list and the f ()-values in another and then using the statistics plot feature on our grapher. Figure 2-1b shows that the graph of f contains all the points in the plot. The concave side of the graph is up. In Problem 1, the rate of growth could be seen as the change in f-values for consecutive -values. Here, the change in f is eponential growth, so the rate of growth is increasing. This parallels algebraicall the graphical concavit at the end of the problem. Problem 2 asks students what graphical evidence distinguishes the power function graph from the eponential function graph in Problem 1. Make sure students realize Figure 2-1b 3. Quadratic Function Problem: In the quadratic function q() ,q() could measure the approimate sales of a new product in the th week since the product was from 0 through 30, and graph function q. Which wa is the concave side of the graph, up or down? What feature does the quadratic function graph have that neither the eponential function graph 4. Linear Function Problem: In the linear function h() 27, h() could equal the number of cents ou pa for a telephone call of length, in from 0 through 18, and graph function h. What does the fact that the graph is neither concave up nor concave down tell ou about the cents per minute ou pa for the call? f () f ()-intercept Concave up Section 2-1: 6 that the graph of the power function passes through the origin, whereas the graph of the eponential function does not. If ou graph the two functions on the same aes, students will see that as gets bigger, the eponential function grows faster than the power function. This is an important distinction between the two tpes of increasing functions. In addition, ou ma want to discuss these questions with our class. a. Based on this mathematical model, how much would ou epect a -ft snake to weigh? b. Wh does the domain of g contain onl nonnegative numbers? c. The equation g () has an eponent in it. Wh do ou suppose it is called a power function rather than an eponential function? In Problem 3, it is important for students to recognize that the quadratic function, unlike an eponential function, has a verte. In addition, ou ma want to discuss these questions with our class. a. The quadratic function has the square of in it. What other geometrical term do ou suppose gives quadratic functions their name? b. Describe verball what the quadratic model indicates about the pattern through time in the sales of this product. c. How do ou interpret the fact that the quadratic model gives decimal answers for the number of items sold, even though the actual number of items sold must be an integer? Problem 4 presents a linear function. The graph of a linear function is neither concave up nor concave down. Additional CAS Problems 1. Eplain algebraicall wh the change in weight in Problem 2 is a quadratic function when g () is cubic. 2. For f () 0.2 2, calculate f ( 1 1) 2 f () using our CAS. Verif b hand that the CAS results are not in error. See page 981 for answers to Problems 1 4 and CAS Problems 1 and 2. Section 2-1: Shapes of Function Graphs 6

6 Section 2-2 Class Time 1 2 das PLANNING Homework Assignment Da 1: RA, Q1 Q, Problems 1 2 odd Da 2: Problems 2 even, even 2-2 Objective Identifing Functions from Graphical Patterns One wa to tell what tpe of function fits a set of points is b recognizing the properties of the graph of the function. Given the graph of a function, know whether the function is eponential, power, quadratic, or linear and find the particular equation algebraicall. Here is a brief review of the basic functions used in modeling. Some of these appeared in Chapter 1. Teaching Resources Eploration 2-2: Graphical Patterns in Functions Supplementar Problems Technolog Resources Eploration 2-2: Graphical Patterns in Functions TEACHING Important Terms and Concepts Slope-intercept form Point-slope form Slope Verte Verte form Parabola Proportionalit constant Directl proportional Inversel proportional Base- eponential function Natural (base-e) eponential function Section Notes In Section 2-1, students plotted equations of eponential, power, quadratic, and linear functions and eplored the features of the graphs. In this section, the are given graphs, and the use what the learned in Section 2-1 to identif the tpe of function each graph represents. Once the identif the tpe of function, the use information about points on the graph to find a particular equation. 66 Chapter 2: Parent function Linear and Constant Functions General equation: a b (often written m b), where a (or m) and b stand for constants and the domain is all real numbers. This equation is in the slope-intercept form because a (or m) gives the slope and b gives the -intercept. If a 0, then b; this is a constant function. Parent function: Section 2-2 summarizes the graphs and equations of linear, quadratic, power, and eponential functions. Here is more detailed information about each tpe of function. Linear and Constant Functions The name linear comes from the fact that the graph of a linear function is a straight line. Linear functions have no concavit and a constant slope. Transformed function: 1 a 1, called the point-slope form because the graph contains the point 1, 1 and has slope a. The slope, a, is the vertical dilation; 1 is the vertical translation; and 1 is the horizontal translation. Note that point-slope form can also be written 1 a 1, where the coordinates of the fied point 1, 1 both appear with a sign. The form 1 a 1 epresses eplicitl in terms of and thus is easier to enter into our grapher. Graphical properties: The graph is a straight line. The parent function is shown in the left graph of Figure 2-2a, the slope-intercept form is shown in the middle graph, and the point-slope form is shown in the right graph. Verball: For slope-intercept form: Start at b on the -ais, run, and rise a. For point-slope form: Start at 1, 1, run 1, and rise a 1. -intercept 8 Run Rise (negative) Point (, ) Figure 2-2a: Linear functions Point (, ) Run Rise Point (1, 4) 4 2 ( 1) 3 The slope-intercept form of a linear function is given as a 1 b. In this form, a is the slope of the line and b is the -intercept. Man tets use m rather than a to represent the slope. The letter m comes from the French montant, (as in mountain ), meaning the rise. If a 0, the function a 1 b becomes the constant function b. The graph of a constant function is a horizontal line. 66 Chapter 2: Properties of Elementar Functions

7 The eruption of Arenal, an active volcano in Costa Rica. The lava particles follow a parabolic path due to gravitational force. Parent quadratic function 2 Quadratic Functions General equation: a 2 b c, where a 0; a, b, and c stand for constants; and the domain is all real numbers Parent function: 2, where the verte is at the origin Transformed function: k a( h ) 2, called the verte form, with verte at (h, k). The value k is the vertical translation, h is the horizontal translation, and a is the vertical dilation. Verte form can also be written k a( h ) 2, but epressing eplicitl in terms of makes the equation easier to enter into our grapher. Graphical properties: The graph is a parabola (Greek for along the path of a ball ), as shown in Figure 2-2b. The graph is concave up if a 0 and concave down if a 0. Power Functions 3 0.4( 1) 2 Figure 2-2b: Quadratic functions General equation: a b, where a and b stand for nonzero constants. If b 0, then the domain can be all real numbers. If b 0, then the domain ecludes 0 to avoid division b zero. If b is not an integer, then the domain usuall ecludes negative numbers to avoid roots of negative numbers. The domain is also restricted to nonnegative numbers in most applications. Parent function: b Verball: For a b, if b 0, then varies directl with the bth power of, or is directl proportional to the bth power of ; if b 0, then varies inversel with the bth power of, or is inversel proportional to the bth power of. The dilation factor a is the proportionalit constant. Translated function: d a( c ) b, where c and d are the horizontal and vertical translation, respectivel. Compare the translated form with 1 a 1 for linear functions k a( h ) 2 for quadratic functions Unless otherwise stated, power function will impl the untranslated form, a b. Section 2-2: If a 0 and b 0, the function is of the form a, which represents a line through the origin. If this function is translated so the point at the origin moves to the point ( 1, 1 ), then the equation becomes 1 1 a( 2 1 ). This form is called point-slope form because it contains both the coordinates of a point on the graph and the slope. It is important for students to master the point-slope form of a line. Verte (1, 3) 3 2( 1) 2 67 This form is more general than the slopeintercept form because an point can be used for ( 1, 1 ), whereas the slope-intercept calls for a specific point, the -intercept. Verte (1, 3) Quadratic Functions A quadratic function is a second-degree polnomial function (i.e., a polnomial function in which the highest power of is 2). The graph of a quadratic function is called a parabola. The left graph in Figure 2-2b shows the parent quadratic function, 2. An other quadratic function is a transformation of this function. The middle graph shows ( 2 1) 2, which is a dilation of the parent graph b a factor of 0.4 and a translation of 1 unit horizontall and 3 units verticall. The graph on the right shows ( 2 1) 2, which has the same translations but a dilation factor of 22. If the vertical dilation is positive, the parabola opens up and is said to be concave up (middle, Figure 2-2b). If the vertical dilation is negative, the parabola opens down and is said to be concave down (right, Figure 2-2b). The equation ( 2 1) 2 can be written in polnomial form ( ) Epand the squared binomial Distribute the 22 and combine like terms. This equation is now in the polnomial form a 2 1 b 1 c. When a quadratic function is written in this form, a positive value of a indicates that the parabola is concave up and a negative value of a indicates that the parabola is concave down. The -intercept is c. The polnomial form a 2 1 b 1 c does not tell us as much about the graph of the quadratic function as does the verte form k 1 a ( 2 h) 2. Section 2-2: Identifing Functions from Graphical Patterns 67

8 Section Notes (continued) Power Functions The power function has the general form of a b. Figure 2-2c shows graphs of three power functions. The graph of contains the origin because 0 when 0. The graph is concave up because the -values increase more and more rapidl as increases. The graph of contains the origin, is increasing, and is concave down because the -values increase at a slower and slower rate as increases. The graph of 3 21 is undefined when 0 (because _ ) so it has no -intercept. The -values decrease as increases because ou are dividing b larger and larger numbers. This list summarizes the features of graphs of functions of the form a b. If a. 0 and b. 1, then the graph contains the origin, is increasing, and is concave up in the first quadrant. If a. 0 and 0, b, 1, then the graph contains the origin, is increasing, and is concave down in the first quadrant. If a. 0 and b, 0, then the graph does not contain the origin and is decreasing and asmptotic to both aes in the first quadrant. If b 1, then the function is a linear function. If b is a non-integer, then the domain is 0 because of the ambiguit of the solution. For eample, (28) 1/3 22, but (28) 2/6 2. Marie Curie was awarded the Nobel Prize in chemistr for the discover of radioactive elements (polonium and radium) in The breakdown of radioactive elements follows an eponential function. 68 Chapter 2: Graphical properties: Figure 2-2c shows power function graphs for different values of b. In all three cases, a 0. The shape and concavit of the graph depend on the value of b. The graph contains the origin if b 0; it has the aes as asmptotes if b 0. The function is increasing if b 0; it is decreasing if b 0. The graph is concave up if b 1 or if b 0 and concave down if 0 b 1. The concavit of the graph describes the rate at which increases. For b 0, concave up means is increasing at an increasing rate, and concave down means is increasing at a decreasing rate. Increasing Increasing Decreasing Concave up Origin Eponent greater than 1 Eponential Functions Origin Eponent between 0 and 1 Figure 2-2c: Power functions General equation: a b, where a and b are constants, a 0, b 0, b 1, and the domain is all real numbers Parent function: b, where the asmptote is the -ais Verball: In the equation a b, varies eponentiall with. Translated function: a b c, where the asmptote is the line c. Unless otherwise stated, eponential function will impl the untranslated form, a b. Graphical properties: Figure 2-2d shows eponential functions for different values of a and b. The constant a is the -intercept. The function is increasing if b 1 and decreasing if 0 b 1 (provided a 0). If a 0, the opposite is true. The graph is concave up if a 0 and concave down if a 0. Increasing -intercept value of a Concave up -intercept value of a Base greater than 1 Concave down Decreasing Concave up Base between 0 and 1 Figure 2-2d: Eponential functions Concave up Origin 3 1 Eponent negative a 68 Chapter 2: Properties of Elementar Functions

9 Mathematicians usuall use one of two particular constants for the base of an eponential function: either, which is the base of the decimal sstem, or the naturall occurring number e, which equals To make the equation more general, multipl the variable in the eponent b a constant. The (untranslated) general equations are given in the bo. DEFINITION: Special Eponential Functions a 1 0 b a e b base- eponential function natural (base-e) eponential function where a and b are constants and the domain is all real numbers. Note: The equations of these two functions can be generalized b incorporating translations in the - and -directions. You ll get a b(c) d and a e b(c) d. Base-e eponential functions have an advantage when ou stud calculus because the rate of change of e is equal to e. In this eploration, ou ll find the particular equation of a linear, quadratic, power, or eponential function from a given graph. EXPLORATION 2-2: Graphical Patterns in Functions 1. Identif what kind of function is graphed, 4. What graphical evidence do ou have that the and find its particular equation. function graphed is an eponential function, not a power function? Find its particular equation (3, 2) (8, 13) 2. Does our graph agree with the given one? 3. concave down, or neither? 1 Section 2-2: Eploration Notes Eploration 2-2 provides practice in finding the particular equation of a linear, quadratic, power, or eponential function from a given graph. Allow students 20 2 minutes to complete this activit. 1. Linear The answer checks. 3. Neither (3, 4) (8, 9). Does our graph agree with the given one? 6. concave down, or neither? continued The graph is not a power function because the -intercept is not zero ( ). The answer checks. 6. Concave up 1 Eponential Functions An eponential function has equation a b, where a 0, b. 0, and b 1. Figure 2-2d on page 68 shows graphs of three eponential functions. In the function , the base is greater than 1. This causes the -values to increase as increases. The base in the function is greater than 0 but less then 1. Raising such a number to greater and greater powers gives smaller and smaller results, so the -values decrease as increases. Both graphs are concave up the left graph because increases at a faster and faster rate and the middle graph because decreases at a slower and slower rate. The graph shows what can happen if a is negative. This list summarizes the features of graphs of functions of the form a b, where a 0, b. 0, and b 1. If a. 0 and b. 1, then the graph is increasing and concave up. The graph is asmptotic to the -ais and has -intercept a. If a. 0 and 0, b, 1, then the graph is decreasing and concave up. The graph is asmptotic to the -ais and has -intercept a. If a, 0 and b. 1, then the graph is decreasing and concave down. The graph is asmptotic to the -ais and has -intercept a. If a, 0 and 0, b, 1, then the graph is increasing and concave down. The graph is asmptotic to the -ais and has -intercept a. Notice that in all four cases the graph is asmptotic to the -ais and has -intercept a. Section 2-2: Identifing Functions from Graphical Patterns 69

10 Section Notes (continued) In the eamples, students identif the tpe of function from a given graph and then find the equation. The function in Eample 1 is linear. Part d shows how to find the equation b writing and solving a sstem of equations. The sstem is solved b the elimination method, but if our students have studied matrices, ou can present the alternative solution given at the end of the eample. (Students will stud matrices in Chapter 13.) The equation can also be found b calculating the slope and then writing the equation in point-slope form: slope So, using the point (, 19), the equation is ( 2 ) EXPLORATION, continued 7. What graphical evidence do ou have that this function is a power function, not an eponential function? Find its particular equation EXAMPLE 1 f() (, 19) (, 6) 1 Figure 2-2e (3, 1) SOLUTION (8, 2) 8. Does our graph agree with the given one? 9. concave down, or neither? 1 For the function graphed in Figure 2-2e, a. Identif the kind of function it is.. Identif what kind of function is graphed, and find its particular equation b. On what interval or intervals is the function increasing or decreasing? Which wa is the graph concave, up or down? c. From our eperience, describe something in the real world that a function with this shape graph could model. d. Find the particular equation of the function, given that points (, 19) and (, 6) are on the graph. e. Confirm b plotting that our equation gives the graph in Figure 2-2e. a. Because the graph is a straight line, the function is linear. (3, 26.6) (8, 38.6) (14, 13.4) 11. Does our graph agree with the given one? 12. concave down, or neither? 13. What did ou learn as a result of doing this eploration that ou did not know before? b. The function is decreasing over its entire domain, and the graph is not concave in either direction. c. The function could model anthing that decreases at a constant rate. The histor tet ou have left to read as a function of the number of minutes ou have been reading. d. f () a b Write the general equation. Use f () as shown on the graph, and use a for the slope. 70 Chapter 2: 7. The -ais appears to be a vertical asmptote, which indicates a power function with a negative eponent The answer checks. 9. Concave up. Quadratic; The answer checks. 12. Concave down 13. Answers will var. 70 Chapter 2: Properties of Elementar Functions

11 f() Figure 2-2f EXAMPLE 2 19 a b 6 a b Section 2-2: Substitute the given values of and into the equation of f. 13 a a 2.6 Subtract the first equation from the second to eliminate b. 6 (2.6) b b 32 Substitute 2.6 for a in one of the equations. f () Write the particular equation. e. Figure 2-2f shows the graph of f, which agrees with the given graph. Note that the calculated slope, 2.6, is negative, which corresponds to the fact that f () decreases as increases. matrices. a b 19 The given sstem. a b a b a b a 2.6 and b 32 Write the sstem in matri form. Multipl both sides b the inverse matri. Complete the matri multiplication. You ll stud the matri solution of linear sstems more full in Section For the function graphed in Figure 2-2g, a. Identif the kind of function it could be. b. On what interval or intervals is the function increasing or decreasing? Which wa is the graph concave, up or down? c. Describe something in the real world that a function with this shape graph could model. d. Find the particular equation of the function, given that points (1, 76), e. Confirm b plotting that our equation gives the graph in Figure 2-2g. 0 0 (3, 94) (2, 89) (1, 76) Figure 2-2g 71 The quadratic equation in Eample 2 is found b writing and solving a sstem of three equations in three variables. Part d shows how to find the solution using matrices. While this eample is eas to do algebraicall, most sstems of three equations in three variables are not eas to solve without matrices. If our students have not learned about matrices, ou can present this alternative algebraic solution: 76 a 1 b 1 c 89 4a 1 2b 1 c 94 9a 1 3b 1 c Subtract the first equation from the second and subtract the second equation from the third to eliminate c and get two equations in two variables: 13 3a 1 b a 1 b Now solve this sstem of two equations to find a and b: 13 2 (3a 1 b) 2 (a 1 b) Subtract the second equation from the first. 8 22a Simplif. a 24 Solve for a. 13 3(24) 1 b Substitute 4 for a in the first equation. b 2 Solve for b. To find c, go back to one of the original equations c Substitute 4 for a and 2 for b in the original first equation. c Solve for c. Section 2-2: Identifing Functions from Graphical Patterns 71

12 Section Notes (continued) The curve in Eample 3 represents a power function. The shape of the curve is similar to that of an eponential function, but it can be distin guished from an eponential function graph because it passes through the origin. Finding the particular equation for the curve requires the use of logarithms. If students are not familiar with logarithms, ou can simpl give them the equation and eplain that the will learn how to find the equation themselves in Section 2-4. The curve in Eample 4 on page 73 represents an eponential function. You ma need to remind students that if the plan to use the result of a calculation in subsequent calculations, the should use the unrounded value. One wa to do this is to store the result in their grapher s memor and then use the stored value rather than the rounded value in the calculation. You ma need to eplain how to store values. Eplain to students that an ellipsis is used to indicate that the digits of a number continue after the last digit displaed. A rounded answer should be preceded with an approimatel equal to sign () to indicate that the answer is an approimation. When a problem does not specif how to round a number, students should round appropriatel, depending on the real-world contet of the problem. SOLUTION EXAMPLE 3 (6, 11.2) 0 0 (4, 44.8) Figure 2-2h SOLUTION a. The function could be quadratic because it has a verte. b. The function is increasing for 3 and decreasing for 3, and it is concave down. c. The function could model anthing that rises to a maimum and then falls back down again, such as the height of a ball as a function of time or the grade ou could earn on a final eam as a function of how long ou stud for it. (Cramming too long might lower our score because of our being sleep from staing up late!) d. a 2 b c Write the general equation. 76 a b c 89 a 2b c 9a 3b c Substitute the given - and -values. Solve b matrices. Write the equation. e. value of a is negative, which corresponds to the fact that the graph is concave down. For the function graphed in Figure 2-2h, a. Identif the kind of function it could be. b. On what interval or intervals is the function increasing or decreasing? Which wa is the graph concave, up or down? c. Describe something in the real world that a function with this shape graph could model. d. Find the particular equation of the function ou identified in part a, given e. Confirm b plotting that our equation gives the graph in Figure 2-2h. a. The function could be a power function or an eponential function, but a power function is chosen because the graph appears to contain the origin. in the -direction. b. The function is increasing and concave up over the entire domain shown. c. The function could model anthing that starts at zero and increases at an increasing rate, such as the power generated b a windmill as a function of wind speed, when the driver applies the brakes, or the volumes of geometricall similar objects as a function of their lengths. Differentiating Instruction Make sure ELL students understand that verte is singular and vertices is plural. In man languages, vertice is the singular. The same is true about matri and matrices. Some ELL students ma not have learned logarithms. Teach them enough to understand Eample 3. Make sure ELL students know how to use STORE on their grapher. 72 Chapter 2: Make sure ELL students understand rise over run. Slope is not referred to in this wa in other languages the standard reference is change in over change in. Some students ma avoid using pointslope form because it appears to be more difficult. Point out the advantages of point-slope form over slope-intercept form. Students will benefit from drawing samples of the various functions and their equations in their journals, so the can refer to specific eamples. Consider allowing ELL students to do the Reading Analsis questions in pairs. The will need an etended transition to doing them alone. Students should enter Problems 1 8 in their journals. The ma also benefit from doing these problems in pairs. 72 Chapter 2: Properties of Elementar Functions

13 d. a b Write the untranslated general equation. Technolog Notes a b 11.2 a 6 b 11.2 b a 6 a b Substitute the given - and -values into the equation. Divide the second equation b the first to eliminate a b Th e a s cancel, and 6 b 6_ b b 1. b. log 3.37 log 1. b Take the logarithm of both sides to get b out of Eploration 2-2 asks students to identif what kinds of functions represent given graphs and to write equations for the graphs. The highresolution graphing capabilities of either Fathom or Sketchpad would be useful for this eploration. 20 EXAMPLE 4 (2, ) (, 6) Figure 2-2i 1 SOLUTION log 3.37 b log 1. log 3.37 b log 1. 3 a 3 Substitute 3 for b in one of the equations. a Write the particular equation. e. value of b is between 0 and 1, which corresponds to the fact that the graph is concave up. For the function graphed in Figure 2-2i, a. Identif the kind of function it could be. b. On what interval or intervals is the function increasing or decreasing? Which wa is the graph concave, up or down? c. Describe something in the real world that a function with this shape graph could model. d. Find the particular equation of the function, given that the points (2, ) and (, 6) are on the graph. e. Confirm b plotting that our equation gives the graph in Figure 2-2i. a. The function could be eponential or quadratic, but eponential is chosen because the graph appears to approach the -ais asmptoticall. CAS Suggestions A CAS does not instantl produce the transformed function form of a given general equation of a quadratic function. Students must know the algebraic technique needed to displa the desired results. Allowing students to use a CAS provides instant feedback to the students as the appl the techniques the are learning. Using a CAS to solve sstems of equations associated with points on a curve allows differentiation between problems intended to assess algebraic manipulation skills from problems intended to determine if students can identif the proper form of a curve from its graph. Consider asking students to compute equations of linear, eponential, or power functions given an two points in Quadrant I. b. The function is decreasing and concave up over its entire domain. c. The function could model an situation in which a variable quantit starts at some nonzero value and decreases, graduall approaching zero, such as the number of degrees a cup of coffee is above room temperature as a function of time since it started cooling. Section 2-2: For Problems 21 24, ELL students ma do better if the write the equation before sketching the graph and answering the questions. You ma need to help them epress their answers, as the require students to eplain. 73 Section 2-2: Identifing Functions from Graphical Patterns 73

14 PROBLEM NOTES Supplementar Problems for this section are available at keonline. Be sure to discuss all the assigned problems, because the contain ke concepts important for students to master and understand. Q1. 9 Q2. 0 Q3. 9 Q4. 8 Q. 1 Q Q7. Q8. 0 In Problem Q9, some students ma recognize that h(29) 3i. Q9. Undefined Q. D Problems 1 7 help students review the important features of the graphs and equations of the various tpes of functions. 1. In power functions, the eponent is constant and the independent variable is in the base. In eponential functions, the base is constant and the independent variable is in the eponent. 2. Quadratic functions have either a maimum or a minimum point. Eponential, linear, and man power functions do not have these. 3. Answers will var. 4. Direct variation power functions have the form a n with n. 0, so 0 when 0. But inverse variation power functions are undefined at Eclude straight lines from being called quadratic 7. (264) 1/2 is undefined but (264) 1/3 24. The restriction allows the function to be defined for all values of. Problem Set 2-2 Reading Analsis 74 Chapter 2: d. a b Write the untranslated general equation. a b 2 6 a b 6 ab ab b 3 Substitute the given - and -values. Divide the second equation b the first to eliminate a /3 b Raise both sides to the 1_ power to eliminate the 3 eponent of b. b Store without rounding. a 2 b in one of the equations. a From what ou have read in this section, what do ou consider to be the main idea? What is the difference between the parent quadratic function and an other quadratic function? How does the -intercept of an eponential function differ from the -intercept of a power function? Sketch the graph of a function that is increasing but concave down. min Quick Review Q1. If f () 2, find f (3). Q2. If f () 2, find f (0). Q3. If f () 2, find f (3). Q4. If g () 2, find g (3). Q. If g () 2, find g (0). Q6. If g () 2, find g (3). Q7. If h() 1/2, find h(2). Q8. If h() 1/2, find h(0). Q9. If h() 1/2, find h(9). Store without rounding. Write the particular equation. e. value of b is between 0 and 1, which corresponds to the fact that the function is decreasing. Q. What propert of real numbers is illustrated b 3( ) 3( )? A. Associative propert of multiplication B. Commutative propert of multiplication C. Associative propert of addition D. Commutative propert of addition E. Distributive propert of multiplication over addition 1. both have eponents. What major algebraic difference distinguishes these two tpes of functions? 2. What graphical feature do quadratic functions have that linear, eponential, and power functions do not have? 3. Write a sentence or two giving the origin of the word concave and eplaining how the word applies to graphs of functions. 74 Chapter 2: Properties of Elementar Functions

15 4. contain the origin but inverse variation power functions do not.. reciprocal function f () 1_ is also a power function. 6. In the definition of quadratic function, what is the reason for the restriction a 0? 7. The definition of eponential function, a b, includes the restriction b 0. Suppose that (). What would equal if were 1_ If were 1_ 3? Wh do ou think there is the restriction b 0 for eponential functions? 8. The verte form of the quadratic-function equation can be written as k a( h ) 2 or k a( h ) 2 are plotting graphs on our grapher and wh the second form is more useful for understanding the translations involved. 9. Reading Problem: Clara has been reading her histor assignment for 20 min and is now starting page 6 in the tet. She reads at a (relativel) constant rate of 0.6 page per minute. a. Find the particular equation epressing the page number she is on as a function of minutes, using the point-slope form. Transform our answer to the slope-intercept form. b. Which page was Clara on when she started reading the assignment? c. The assignment ends at the top of page 63. When would ou epect Clara to finish?. Baseball Problem: Ruth hits a high fl ball to when she hits it. Three seconds later it reaches its a. Write an equation in verte form of the quadratic function epressing the height of the ball eplicitl as a function of time. 2? b. How high is the ball s after it was hit? c. If nobod catches the ball, how man seconds after it was hit will it reach the ground? function graph is shown. a. Identif the tpe of function it could represent. b. On what interval or intervals is the function increasing or decreasing and which wa is the graph concave? c. From our eperience, what relationship in the real world could be modeled b a function with this shape graph? d. Find the particular equation of the function if the given points are on the graph. e. Confirm b plotting that our equation gives the graph shown (6, ) Section 2-2: 20 (0, 20) (1, 16) (4, 1) 9 (1, 6) (3, 4) (, 18) 7 Problem 8 helps students recognize that there are advantages and disadvantages to both forms of the quadratic-function equation. 8. The grapher onl allows ou to enter equations in form. The second form shows the horizontal translation h and the vertical translation k. Problems 9 and require students to find equation models for situations based on written descriptions of the situations. Students ma use the greatest integer function in Problem 9, because the function is being used to describe the page number she is on, which is a discreet function. 9a ( 2 20) b. Page 44 9c. 11 2_ 3 min from now. a ( 2 3) 2 b. 84 ft c s Problems are trivial if students use a CAS to solve sstems. 11a. Linear 11b. Increasing for all real-number values of, not concave 11c. Answers will var. 11d e. The graphs match. 12a. Linear 12b. Decreasing for all real-number values of, not concave 12c. Answers will var. 12d e. The graphs match. 13a. Quadratic 13b. Decreasing for, 2.2 and increasing for. 2.2, concave up 13c. Answers will var. 13d e. The graphs match. Section 2-2: Identifing Functions from Graphical Patterns 7

16 Problem Notes (continued) 14a. Quadratic 14b. Increasing for, 4 and decreasing for. 4, concave down 14c. Answers will var. 14d e. The graphs match. 1a. Eponential 1b. Increasing for all real-number values of, concave up 1c. Answers will var. 1d. (1.3) 1e. The graphs match. 16a. Eponential 16b. Decreasing for all real-number values of, concave up 16c. Answers will var. 16d. 96 (0.) 16e. The graphs match. 17a. Power 17b. Increasing for 0, concave down 17c. Answers will var. 17d. lo g e. The graphs match. 18a. Power (inverse) 18b. Decreasing for 0, concave up 18c. Answers will var. 18d e. The graphs match. Problems help students review the meanings of directl proportional, inverse variation, and direct variation. Problem 2 gives students a natural eponential function and asks them to find an equivalent eponential function with a different base. 2. b e The graphs are equivalent. See pages for answers to Problems and CAS Problems (4, 30.6) 30 (2, 28.2) (1, 2.2) (2, 8.4) (1, 6.) (1, 48) (2, 24) 4 (2, 8) (1, ) (2, 6) (, 2.4) (1, 3) (0, 0) (4, 24) 76 Chapter 2: (, 8) Additional CAS Problems 1. Given the general equation of an particular linear function, use a CAS to move the constant to the other side of the equation, add a nonzero multiple of the slope to both sides, and factor the result. What is the form of the result? What new information does it give? Repeat the process with a different nonzero multiple of the slope. What new 21. Suppose that increases eponentiall with and that z is directl proportional to the square of. Sketch the graph of each tpe of function. In what was are the two graphs similar to each other? What major graphical difference would allow ou to tell which graph is which if the were not labeled? 22. Suppose that decreases eponentiall with and that z varies inversel with. Sketch the graph of each tpe of function. Give at least three was in which the two graphs are similar to each other. What major graphical difference would allow ou to tell which graph is which if the were not labeled? 23. Suppose that varies directl with and that z increases linearl with. direct variation function is a linear function but a linear function is not necessaril a direct variation function. 24. Suppose that varies directl with the square of and that z is a quadratic function of. wh the direct-square variation function is a quadratic function but the quadratic function is not necessaril a direct-square variation function. 2. Natural Eponential Function Problem: Figure 2-2j shows the graph of the natural eponential function f () 3 e 0.8. g () 3 b. Find the value of b for which g () f (). Show graphicall that the two functions are equivalent Figure 2-2j 2 information does this result give? How can ou use this information to confirm the graph of the original linear function? 2. Find an equation of an eponential function containing the points (2, 7) and (7, 13). 3. Is the power function that contains the points ( 3, 192) and 4, 27 4 even, odd, or neither? 76 Chapter 2: Properties of Elementar Functions

17 2-3 Objective Identifing Functions from Numerical Patterns A 16-in. pizza has four times as much area as an 8-in. pizza. A grapefruit whose diameter is cm has eight times the volume of a grapefruit with diameter cm. In general, when ou double the linear dimensions of a three-dimensional object, ou multipl the surface area b time ou add 00 mi to the distance ou have driven our car, ou add a constant amount sa, $300 to the cost of operating that car. In this section ou will use such patterns to identif the tpe of function that fits a given set of function values. Then ou will find more function values, either b following the pattern or b finding the equation of the function. -values and the corresponding -values, identif which tpe of function the fit (linear, quadratic, power, or eponential). equation. The Add Add Pattern of Linear Functions f () Linear function f() Th e add-add propert Figure 2-3a Figure 2-3a shows the graph of the linear function f () As ou can see from the graph and the adjacent table, each time ou add 2 to, increases b 3. This pattern emerges because a linear function has constant slope. Verball, ou can epress this propert b saing that ever time ou add a constant to, ou add a constant (not necessaril the same as the constant added to ) to. This propert is called the add add propert of linear functions. Section 2-3 Class Time 2 das PLANNING Homework Assignment Da 1: RA, Q1 Q, Problems 1 23 odd Da 2: Problems 2 27, 29 32, 3 Teaching Resources Eploration 2-3: Patterns for Quadratic Functions Eploration 2-3a: Numerical Patterns in Function Values Eploration 2-3b: Equations from Given Values Practice Supplementar Problems Test 4, Sections 2-1 to 2-3, Forms A and B Technolog Resources Eploration 2-3a: Numerical Patterns in Function Values Activit: Moore s Law Activit: Population Growth TEACHING Important Terms and Concepts Add add propert Add multipl propert Multipl multipl propert Second differences Discrete data Third differences Quartic function Section 2-3: 77 Section 2-3: Identifing Functions from Numerical Patterns 77

18 g() Eponential function 2 Section Notes In Section 2-2, students learned to recognize the tpe of function based on its graph. In this section, students learn to use numerical patterns in the - and -values to identif the function tpe. It is recommended that ou spend two das on this section. On Da 1, present the add multipl pattern of eponential functions and the multipl multipl pattern of power functions using the section s graphs and tables. You ma prefer to use Eploration 2-3a in the Instructor s Resource Book to introduce these patterns. Have students do Eploration 2-3 on page 79, which introduces the pattern for the quadratic function. Students usuall learn these patterns quickl. Discuss the add add, add multipl, multipl multipl, and seconddifferences patterns, using the graphs and tables given in the section. Be sure to emphasize that to use these patterns to identif function tpe, the -values must be regularl spaced. The bo on page 80 summarizing these four properties should help students as the work on the problems. Eample 1 on page 81 asks students to identif the pattern and then choose the appropriate model, in this case quadratic. Students do not need to find the particular equation for the function. Eample 2 on page 81 gives students two function values and asks them to find a third value in three cases when the function is linear, when it is a power function, and when it is eponential. To find the three values, students must appl the add add, add multipl, and multipl multipl properties, respectivel. This eample helps students to distinguish between the various properties. As in Eample 1, students do not need to find particular equations for the functions g() The add-multipl propert Figure 2-3b 78 Chapter 2: The Add Multipl Pattern of Eponential Functions Figure 2-3b shows the graph of the eponential function g () 3. This time, adding 2 to results in the corresponding g ()-values being multiplied b the constant 9. This is not coincidental. Here s wh the pattern holds. g (1) g (3) (which equals 9 times 1) You can see algebraicall wh this is true. g (3) Write the eponent as 1 increased b ( 3 1 ) 3 2 Associate and 3 1 to get g (1) in the epression. g (1) 9 The conclusion is that if ou add a constant to, the corresponding -value is multiplied b the base raised to that constant. This is called the add multipl propert of eponential functions. The Multipl Multipl Pattern of Power Functions h() Power function Figure 2-3c Figure 2-3c shows the graph of the power function h() 3. As shown in the table, adding a constant, 3, to does not create a corresponding pattern. g() However, a pattern does emerge if ou pick values of that change b being multiplied b a constant. h(3) For Eample 3 on page 82, ou ma need to remind students what varies directl and varies inversel mean. These ideas were discussed in Section 2-2. h(6) (which equals 8 times 13) h(12) Ignore The multipl-multipl propert In Eample 4 on page 82, ou ma need to eplain what a direct-square power function is. Setting up the eample using this table might be helpful. 32 f () 00 20? Chapter 2: Properties of Elementar Functions

19 If ou double the -value from 3 to 6 or from 6 to 12, the corresponding -values are multiplied b 8, or 2 3. You can see algebraicall wh this is true. h(6) Write the -value 6 as twice 3. ( 3 3 ) 2 3 h(3) 8 Distribute the eponent over multiplication and then associate. In conclusion, if ou multipl the -values b 2, the corresponding -values are multiplied b 8. This is called the multipl multipl propert of power in Figure 2-3c. The do belong to the function, but the -values do not fit the multipl pattern. In this eploration, ou ll find patterns for the -values in quadratic functions similar to the addadd propert of linear functions. EXPLORATION 2-3: Patterns for Quadratic Functions 1. Show b making a table on our grapher that the points in the table fit the quadratic function q() On the same screen, plot the graph of q() ( ) and the five data points. You ma use the stat feature on our grapher. Sketch the result.. Trace the graph of q() to each value of in the table. Do the five points lie on the graph? q() 6. Show that a quadratic function fits the data in this table b finding second differences. Find 12.0 the particular equation, and show that these values satisf the equation Find the differences between consecutive -values. Then find the second differences, that is, the differences between the consecutive differences. What do ou notice? 3. Recall that the general equation of a quadratic function is a 2 b c, where a, b, and c stand for constants. Substitute the first three to get three linear equations involving a, b, and c. Solve this sstem of equations using matrices. Write the particular equation. Does f() What did ou learn as a result of doing this eploration that ou did not know before? Eploration Notes Eploration 2-3 gives students practice identifing the second differences for a quadratic function and finding a particular equation for the function. Problem 2 suggests that students use the matri feature on their graphers to find the equation. If our students do not know how to work with matrices, the can solve the sstem using algebraic methods. You might have students complete this eploration in class, after ou have discussed the add add, add multipl, and multipl multipl patterns. Allow students 20 2 minutes to complete this activit. See page 81 for notes on additional eplorations. 1. The values are the same as in the table shown. 2. First differences: 20.2, 1.4, 3.0, 4.6 Second differences: 1.6, 1.6, 1.6 All the second differences are the same! a 1 2b 1 c a 1 4b 1 c a 1 6b 1 c q() Section 2-3: Eample on page 82 is a radioactive-deca problem. The value of f (12) can be found b etending the add multipl pattern in the table. However, to find the value of f (2), it is necessar to find the particular equation. Part d eplains how to do this. Be sure to discuss the note after Eample, which emphasizes that there are man functions that fit a particular set of points. Problem 30 also illustrates this point Yes, the five points lie on the graph. 6. Second differences: 212.6, 212.6, f () Answers will var Section 2-3: Identifing Functions from Numerical Patterns 79

20 The Constant-Second-Differences Pattern of Quadratic Functions 0 q() Quadratic function 9 f() Constant second differences Figure 2-3d Differentiating Instruction Have students, in pairs, write each of these properties in their own words: add add propert, add multipl propert, multipl multipl propert, constant-second-differences-propert. Monitor students as the do the eploration. If ou want ELL students to use the STAT feature on their grapher, ou ma need to teach them how. For Problems 2 27, ELL students ma not be familiar with the relationship between areas and volumes of similar objects. Proofs as in Problems ma be difficult for ELL students, as the concept of proof varies from language to language. You ma want to dela assigning them or make them etra credit. Figure 2-3d shows the graph of the quadratic function q() Afor linear functions applies to quadratics. For equall spaced -values, the differences between the corresponding -values are equall spaced. Thus the differences between these differences (the second differences) are constant. This constant is equal to 2 ad 2, twice the coefficient of the quadratic term times the square of the difference between the -values. These four properties are summarized in the bo. PROPERTIES: Patterns for Function Values Constant second differences First differences Add Add Propert of Linear Functions If f is a linear function, adding a constant to results in adding a constant to the corresponding f ()-value. That is, if f () a b and 2 c 1, then f 2 ac f 1 Add Multipl Propert of Eponential Functions If f is an eponential function, adding a constant to results in multipling the corresponding f ()-value b a constant. That is, if f () a b and 2 c 1, then f 2 b c f 1 Multipl Multipl Propert of Power Functions If f is a power function, multipling b a constant results in multipling the corresponding f ()-value b a constant. That is, if f () a b and 2 c 1, then f 2 c b f 1 Constant-Second-Differences Propert of Quadratic Functions If f is a quadratic function, f () a 2 b c, and the -values are spaced d units apart, then the second differences between the f ()-values are constant and equal to 2ad Chapter 2: 80 Chapter 2: Properties of Elementar Functions

21 EXAMPLE 1 Identif the pattern in these function values and the kind f() of function that has this pattern. SOLUTION f() the constant-second-differences propert, as 1 7 shown in the table. Therefore, a quadratic 1 function fits the data EXAMPLE 2 SOLUTION If function f has values f () 12 and f () 18, find f (20) if f is a. A linear function b. A power function c. An eponential function a. first -value to get the second one and that ou add 6 to the first f ()-value to get the second one. Make a table of values ending at 20. The answer is f (20) 30. f () 24 f () Linear b. the first to the second - and f ()-values, notice that ou multipl b 2 to get and that ou multipl 12 b 1. to get 18. Make a table of values ending at 20. The answer is f (20) Additional Eploration Notes Eploration 2-3a provides practice in finding a pattern in the -values for regularl spaced -values for eponential, power, or linear functions. You might assign this at the beginning of either Da 1 or Da 2 to help students review what the learned the da before. Or ou might assign this activit as a review before the chapter test. Allow students 20 2 minutes to complete this activit. Eploration 2-3b provides practice in finding a particular equation and using it to find other values when a set of regularl spaced - and -values is given. You might assign this eploration to groups on Da 2 or use it as an etra homework assignment or a review sheet. The problems in the eploration also make ecellent quiz or test questions. Allow students 20 2 minutes to complete this activit. 2 2 f () Section 2-3: 81 Section 2-3: Identifing Functions from Numerical Patterns 81

22 Technolog Notes Eploration 2-3a: Numerical Patterns in Function Values in the Instructor s Resource Book focuses on finite differences. Students might benefit from using Fathom s tools for making tables of finite differences. Activit: Moore s Law in Teaching Mathematics with Fathom has students use Fathom to fit an eponential curve to data on the number of transistors in Intel processors since Students are then asked to determine the truth of Moore s Law based on their findings. This would make an ecellent project to use now and then to revisit during Section 3-4. Allow 3 0 minutes. Activit: Population Growth in Teaching Mathematics with Fathom gives eperience with eponential functions based on numerical patterns. Students begin b fitting an eponential growth model to population data, and then the introduce a crowding effect to obtain a logistic function. You could ask students to do onl the eponential part, or ou could have them do the whole activit to foreshadow Section 2-7. Allow 0 minutes. EXAMPLE 3 SOLUTION EXAMPLE 4 SOLUTION EXAMPLE (h) f () (µci) c. adding to results in multipling the corresponding f ()-value b 1.. Make a table of values ending at 20. The answer is f (20) f() Describe the effect on of doubling if a. varies directl with b. varies inversel with the square of. c. varies directl with the cube of. a. is doubled that is, multiplied b 2 1. b. is multiplied b 1_ that is, multiplied b 2 2. c. is multiplied b 8 that is, multiplied b 2 3. Suppose that f is a direct-square power function and that f () 00. Find f (20). Because f 20 as. Multipling f ()-value b 2 because f is a direct square, so f (20) f ( ) 2 f () ,000 Radioactive Tracer Problem: The compound 18-fluorodeoglucose (18-FDG) is composed of radioactive fluorine (18-F) and a sugar (deoglucose). It is used to trace glucose metabolism in the heart. 18-F has a half-life of about 2 h, which means that at the end of each 2-hour time period, onl half of the 18-F that was there at the beginning of the time period remains. Suppose a dose of f () be the number of microcuries (µci) of 18-FDG that remains over time, in hours, as shown in the table. a. Find the number of microcuries that remains after 12 h. b. Identif the pattern these data points follow. What tpe of function shows this pattern? c. Wh can t ou use the pattern to find f (2)? Eponential f () Chapter 2: 82 Chapter 2: Properties of Elementar Functions

23 f () (µci) SOLUTION Graph fits points. Figure 2-3e f() (µci) (h) This graph also fits the points. (h) Figure 2-3f d. Find a particular equation of f (). Show b plotting that all the f ()-values in the table satisf the equation. e. Use the equation to calculate f (2). Interpret the solution. a. Follow the add pattern in the -values until ou reach 12, and follow the multipl pattern in the corresponding f ()-values. (h) f (12) µci b. c. -values. Th e -values skip over 2, so f (2) cannot be found using the pattern. d. f () a b General equation of an eponential function. a b 2 Substitute an two of the ordered pairs. 2. a b 2. a b Divide the equations. Have the larger eponent in a b 2 the numerator. 0. b 2 Simplif. 0. 1/2 b Raise both sides to the 1_ 2 power. b f () (µci) Section 2-3: Store without rounding. a( ) 2 Substitute the value for b into one of the equations. a Solve for a. f () ( ) Write the particular equation. Figure 2-3e shows the graph of f passing through all four given points. e. f (2) ( ) This means that there was about µci of 18-FDG after 2 h. possible for other functions to fit this set of points, such as the function g () ( ) sin See Chapter for the meaning of the 2 sine function. which also fits the given points, as shown in Figure 2-3f. Deciding which function fits better will depend on the situation ou are modeling. Also, ou can test further to see whether our model is supported b data. For eample, to test the second model, ou could collect measurements over shorter time intervals and see if the data have a wav pattern. 83 CAS Suggestions Numerical pattern properties for various functions can be verified with a CAS without overwhelming students with intermediate algebra. This is where the power of a CAS is most useful. Once students learn how to ask the general algebraic form of the questions, the CAS handles the difficult algebra, and the student is left to interpret and use the results. Encourage students to find was to prove numerical pattern properties on their own using a CAS. Eample 3 can be solved using a CAS b first defining the functions. In this case, doubling in parts b and c multiplies the function b 1_ 4 and 8, respectivel. At this level, students should be able to look at the results in these screens and read from the algebraic form that doubling multiplies the corresponding -values b 1_ 4 and 8, respectivel. If ou want to be more eplicit, enter f (2) to see f () the multipliers in isolation. Section 2-3: Identifing Functions from Numerical Patterns 83

24 PROBLEM NOTES Supplementar Problems for this section are available at keonline. Using a CAS to solve the problems in this problem set should not be seen as disabling their intent. Knowing what question to ask of a CAS and how to ask it is a comple mathematical skill. Q1. a 1 b Q2. a b, a 0, b 0 Q3. a b, b. 0, b 1, a 0 Q4. a 2 1 b 1 c, a 0 Q. Power Q6. Eponential Q7. Vertical dilation b 4 Q8. B Q9. Q. Problems 1 24 follow the eamples and should be routine for students. 1. Add add propert: linear 2. Multipl multipl propert: power, inverse variation 3. Multipl multipl propert: power; and constant-second-differences propert: quadratic Problem Set 2-3 Reading Analsis From what ou have read in this section, what do ou consider to be the main idea? How is the the concept of slope? How do the properties of eponential and power functions differ? If ou triple the diameter of a circle, what effect does this have on the circle s area? What tpe of function has this propert? What numerical pattern do quadratic functions have? min Quick Review Q1. Write the general equation of a linear function. Q2. Write the general equation of a power function. Q3. Write the general equation of an eponential function. Q4. Write the general equation of a quadratic function. Q. f () 3 is the equation of a particular? function. Q6. f () 3 is the equation of a particular? function. Q7. Name the transformation of f () that gives g () f (). Q8. The function g () 3 f (( 6)) is a vertical dilation of function f b a factor of A. 3 B. C. D. 6 E. 6 Q9. Sketch the graph of a linear function with negative slope and positive -intercept. Q. Sketch the graph of an eponential function with base greater than 1. constant-second-differences pattern. Identif the tpe of function that has the pattern. 84 Chapter 2: 4. Add multipl propert: eponential. Add add propert: linear; multipl multipl propert: power 6. Add add propert: linear 7. Multipl multipl propert: power, inverse variation 8. Add add propert: linear 1. f () f () f () f () f () f () f () f () f () f () Add multipl propert: eponential. Multipl multipl propert: power 11. Constant-second-differences propert: quadratic 12. Constant-second-differences propert: quadratic 84 Chapter 2: Properties of Elementar Functions

25 11. f () f () For value if f is a. A linear function b. A power function c. An eponential function 13. Given f (2) and f (6) 20, find f (18). 14. Given f (3) 80 and f (6) 120, find f 1. Given f () 0 and f (20) 90, find f 16. Given f (1) 00 and f (3) 0, find f (9). the other values specified. 17. Given f is a linear function with f (2) 1 and f () 7, find f (8), f (11), and f 18. Given f is a direct-cube power function with f (3) 0.7, find f (6) and f (12). 19. Given that f () varies inversel with the square of and that f () 1296, find f () and f (20). 20. Given that f () varies eponentiall with and that f (1) 0 and f 90, find f (7), f (), and f (16). f () if ou double the value of. 21. Direct-square power function 22. Direct fourth-power function 23. Inverse variation power function 24. Inverse-square variation power function 2. Volume Problem: The volumes of similarl shaped objects are directl proportional to the cube of a linear dimension. Baseball Volleball Figure 2-3g Section 2-3: a. Recall from geometr that the volume, V, of a sphere equals _ 3 r 3, where r is the radius. V _ 3 r 3 shows that the volume of a sphere varies directl with the cube of the radius. If a baseball has volume 200 c m 3, what is the volume of a volleball that has three times the radius of the baseball (Figure 2-3g)? b. King Kong is depicted as having the same proportions as a normal gorilla but as being times as tall. How would his volume (and thus his weight) compare to that of a normal gorilla? If a normal gorilla weighs weigh? Is this surprising? c. A great white shark 20 ft long weighs about of ears ago suggest that there were once great whites 0 ft long. How much would ou epect such a shark to weigh? d. were 1 as tall as normal people. If Gulliver weighed 200 lb, how much would ou epect a Iris Weddell White s illustration Visits Gulliver in Jonathan Swift s Gulliver s Travels. (The Granger Collection, New York) 26. Area Problem: The areas of similarl shaped objects are directl proportional to the square of a linear dimension. a. Give the formula for the area of a circle. square of the radius. b. If a grapefruit has twice the diameter of an orange, how do the areas of their rinds compare? 8 Problems can be solved easil with a CAS sstem solver, after which the additional function values are trivial to compute. 13a. 6 13b c a b c a. 70 1b. 81 1c a b. 16c f(8) 13, f(11) 19, f(14) f(6).6, f(12) f() 324, f(20) f(7) 81, f() 72.9, f(16) Multipl b Multipl b Divide b Divide b 4. Problems 2 27 require students to appl the facts that the volumes of similarl shaped objects are directl proportional to the cube of a linear dimension and that the areas of similarl shaped objects are directl proportional to the square of a linear dimension. 2a. V(r) has the form V ar 3 where a 4_ 3 ; 400 cm 3. 2b. 400,000 lb 200 tons 2c. 00,000 lb 2d. 0.2 lb 26a. A r 2. A(r) has the form A ar 2 where a. 26b. The grapefruit s rind would have four times as much area as that of the orange. Section 2-3: Identifing Functions from Numerical Patterns 8

26 Problem Notes (continued) 26c The proportion of the original length is squared to find the proportion of the original area. 26d. 200 m 2 27a. 16 times more wing area 27b. 64 times heavier 27c. The full-sized plane had four times as much weight per unit of wing area as the model. Problems 28 and 29 are application problems that depend on students being able to write the particular equation that meets specific conditions. 28a. A(2) $12, A(3) $1331, A(4) $ b. After about 7 ears 29a. [H(3) 2 H(2)] 2 [H(2) 2 H(1)] 232 ft; [H(4) 2 H(3)] 2 [H(3) 2 H(2)] 232 ft; [H() 2 H(4)] 2 [H(4) 2 H(3)] 232 ft 29b. H(t) 216 t t 1 ; H(4) 9; H() 29c. H(2.3) ft; going up; the height seems to peak at about t 3 s. 29d s (going up) or s (coming down); there are two solutions to the equation at 0. 29e s; ft 29f s Problem 30 emphasizes the point that more than one function fits a set of data points. 30a. f () 2 3(2 ) b. c. When Gutzon Borglum designed the reliefs he carved into Mount Rushmore in South Dakota, he started with models 1 12 the lengths of the actual reliefs. How does the area of each model compare to the area of each of the final in the linear dimension results in a relativel large decrease in the surface area to be carved. d. Gulliver traveled to Brobdingnag, where people were times as tall as normal people. If Gulliver had 2 m 2 of skin, how much skin surface would ou epect a Brobdingnagian to have? 27. Airplane Weight and Area Problem: In 1896, an airplane he was designing. In 1903, he tried unsuccessfull to fl the full-size airplane. length of the model (Figure 2-3h). Model Actual Figure 2-3h a. The wing area, and thus the lift, of similarl shaped airplanes is directl proportional to the square of the length of each plane. How man times more wing area did the full-size plane have than the model? b. The volume, and thus the weight, of similarl shaped airplanes is directl proportional to the cube of the length. How man times heavier was the full-size plane than the model? 86 Chapter 2: 30c. c. Wh do ou think the model was able to fl but the full-size plane was not? 28. Compound Interest Problem: Mone left in a savings account grows eponentiall with time. Suppose that ou invest $00 and find that a ear later ou have $10 in our account. a. b. In how man ears will our investment double? 29. Archer Problem: Ann Archer shoots an arrow into the air. The table lists its height at various times after she shoots it. Time (s) Height (ft) a. Show that the second differences between consecutive height values in the table are constant. b. Use the first three ordered pairs to find the particular equation of the quadratic function that fits these points. Show that the function contains all of the points. c. Based on the graph ou fit to the points, how high was the arrow at 2.3 s? Was it going up or going down? How do ou tell? d. At what two times was the arrow 0 ft high? How do ou eplain the fact that there were two times? e. When was the arrow at its highest? How high was that? f. At what time did the arrow hit the ground? 30. The Other Function Fit Problem: It is possible for different functions to fit the same set of discrete data points. Suppose that the data in the table have been given. 4 3(4 ) (6 ) (8 ) ( ) f 2 () also fits the data. Man functions can fit a set of discrete data points. 86 Chapter 2: Properties of Elementar Functions

27 f () f() Figure 2-3i a. Show that the function f () 3 2 fits the data, as shown in Figure 2-3i. b. Select radian mode, and then plot f 1 () 3 2 and f 2 () sin 2 (), where sin is the sine function (see Chapter ). Sketch the result. Does the equation of f 2 () also fit the given data? c. Deactivate f 2 () from part b and plot f 3 () 3 2 cos(), where cos is the cosine function (see Chapter ). Sketch the result. What do the results tell ou about fitting functions to discrete data points? 31. Incorrect Point Problem: B considering second differences, show that a quadratic function does not fit the values in this table What would the last -value have to be for a quadratic function to fit the values eactl? 32. Cubic Function Problem: Figure 2-3j shows the graph of the cubic function f () f() Figure 2-3j 8 a. Make a table of values of f () for each integer value of from 1 to 6. b. Show that the third differences between the values of f () are constant. You can calculate the third differences in a time-efficient wa using the list and delta list features of our grapher. If ou do it b pencil and paper, be sure to subtract (value previous value) in each case. c. Make a conjecture about how ou could determine whether a quartic function (fourth degree) fits a set of points. 33. The Add Add Propert Proof Problem: for a linear function, adding a constant to adds a constant to the corresponding value of f (). Do this b showing that if 2 1 c, then f 2 equals a constant plus f 1. Start b writing the equations of f 1 and f 2, and then make the appropriate substitutions and algebraic manipulations. 34. The Multipl Multipl Propert Proof Problem: b a constant multiplies the corresponding value of f () b a constant as well. Do this b showing that if 2 c 1, then f 2 equals a constant times f 1. Start b writing the equations of f 1 and f 2, and then make the appropriate substitutions and algebraic manipulations. 3. The Add Multipl Propert Proof Problem: constant to multiplies the corresponding value of f () b a constant. Do this b showing that if 2 c 1, then f 2 equals a constant times f 1. Start b writing the equations of f 1 and f 2, and then make the appropriate substitutions and algebraic manipulations. 36. The Constant-Second-Differences Propert Proof Problem:f () a 2 b c.d be the constant difference between successive -values. Find f ( d), f ( 2d), and f ( 3d). Simplif. B subtracting consecutive f ()-values, find the three first differences. B subtracting consecutive first differences, show that the two second differences equal the constant 2a d 2. Section 2-3: 31. If (8) were 2, then a quadratic function would fit. Problem 32 etends the concept of second differences and quadratics to third differences and cubics. 32a. f () b. The third differences equal 6. 32c. A quartic function will have constant fourth differences. Problems guide students through proofs of the properties from this section. 33. If f() a 1 b, then f( 2 ) f( 1 1 c) 1 b a 1 1 ac 1 b ( a 1 1 b) 1 ac f( 1 ) 1 ac. 34. If f() ab, then f( 2 ) f( c 1 ) a( c 1 ) b a( c b b 1 ) c b b a 1 c b f( 1 ). 3. If f() a b, then f( 2 ) f(c 1 1 ) ab c1 1 a( b c b 1 ) b c ab 1 b c f( 1 ). Additional CAS Problems 1. What is the effect on each of the following functions of multipling b 2.? a. f () a 4 b. varies inversel with the square root of. c. varies directl with the fifth power of. 2. Cubic functions are constant on the third common difference. If the general form of a cubic function is f () a 3 1 b 2 1 c 1 d and w is the distance between successive -values, then what is the constant value of the third differences in the f () values? (Note: Problem 32 does this for a particular cubic function, but a CAS makes the general solution just as accessible as Problem 32 is without a CAS.) 3. Without using an statistical functions, compute the particular equations of the functions represented b the coordinate pairs in Problems See page 982 for answers to Problem 36 and CAS Problems 1 3. Section 2-3: Identifing Functions from Numerical Patterns 87

28 Section 2-4 Class Time 1 da PLANNING Homework Assignment RA, Q1 Q, Problems 1 47 odd Teaching Resources Eploration 2-4: Introduction to Logarithms Eploration 2-4a: Introduction to Logarithmic Functions Technolog Resources CAS Activit 2-4a: Dilations of Eponential Functions TEACHING Important Terms and Concepts Base- logarithm Logarithm Logarithm of a product Logarithm of a quotient Logarithm of a power A logarithm is an eponent Eploration Notes Eploration 2-4 introduces students to the properties of logarithms and using logarithms to solve eponential equations. Note that TI-Nspire handhelds allow ou to enter a base. If ou do not enter a base, it will automaticall enter base. 1. log 1 log 0 2 log 00 3 log log 2. log means the eponent in the power of that equals. 3. Conjecture: log( 1.8 ) Chapter 2: 2-4 Objective Properties of Logarithms An positive number can be written as a power of. For instance, logarithms of 3,, and 1, respectivel. log 3 log log Learn the properties of base- logarithms. ou an algebraic wa to solve eponential equations such as EXPLORATION 2-4: Introduction to Logarithms 1. Th e LOG ke on our calculator finds the logarithms: 4. Test the value of 1.8 on our calculator and then finding the logarithm of the (unrounded) answer. log?. Test our conjecture again b finding log 0? log 00? log 000? log? 2. From what log means. Write what ou discover. 3. Based of the answer ou get. 6. Find log 2 and log 32. Show numericall that log 32 is five times log Note that Thus, log 2 log 2. Show numericall that log Complete the propert of the log of a power: should log 1.8 equal? log b? log , confirming the conjecture.. log , again confirming the conjecture. 6. log log log 2 ( ) , which equals log What did ou learn as a result of doing this eploration that ou did not know before? 7. log log log ( ) , which equals log log b log b 9. Answers will var. 88 Chapter 2: Properties of Elementar Functions

29 A decibel, which measures the relative intensit of sounds, has a logarithmic scale. Prolonged eposure to noise intensit eceeding 8 decibles can lead to hearing loss. Definition and Properties of Base- Logarithms To gain confidence in the meaning of logarithm, press LOG 3 on our calculator. You will get log 3 Then, without rounding, raise to this power. You will get The powers of have the normal properties of eponentiation. For instance, 1 (3)() Add the eponents; keep the same base You can check b calculator that reall does equal 1. From this eample ou can infer that logarithms have the same properties as eponents. This is not surprising, because logarithms are eponents. For instance, log(3 ) log 3 log Th e log of a product equals the sum of the logs of the factors. From the values given earlier, ou can also show that log 1 log 1 log 3 The log of a quotient. 3 This propert is reasonable because ou divide powers of equal bases b subtracting the eponents Because a power can be written as a product, ou can find the logarithm of a power like this: log( ) log 3 log 3 log 3 log 3 Combine like terms. The logarithm of a power equals the eponent of that power times the logarithm of the base. To verif this result, observe that LOG 3 on our LOG 81. You get the same answer, Section Notes When discussing the definition and properties of logarithms, the most important idea to stress is that a logarithm is an eponent, so logarithms obe the same laws eponents do. Refer to the Additional Class Eamples, which can help students see the connection between eponents and logarithms. Eamples 1 and 2 on pages 90 and 91 are direct applications of the definition of logarithm. Eamples 3 and 7 on pages 91 and 92 appl the properties of logarithms, and Eample 6 on page 92 gives the proof of the logarithm of a product propert. As ou work through the eamples with students, keep coming back to the idea that a logarithm is an eponent. This section discusses onl base- logarithms, which are also called common logarithms. The net section introduces base-e logarithms, which are called natural logarithms, as well as the change-of-base propert. If our students use TI-Nspire, consider asking them to enter different values for b as the base of a logarithm. What appears to be the range of acceptable values for b to determine real values for a logarithm? Eplain wh this restriction is reasonable. Section 2-4: 89 Additional Eploration Notes Eploration 2-4a enforces the notion that logarithms are eponents b comparing base- logarithms with powers of. It finishes b considering other bases, so it makes a good transitional activit from Section 2-4 to Section 2-. Section 2-4: Properties of Logarithms 89

30 Differentiating Instruction If ou assign Problems 4 and 46, ou ma need to help ELL students with the proof. ELL students ma struggle with the language in Problem 47. Have students put the base- logarithmic properties in their journal, both smbolicall and verball, for ELL students, perhaps in their primar language. Verif that ELL students have done the Reading Analsis questions, perhaps in pairs. Slide rules, used b engineers in the 19th and earl 20th centuries, emplo the principle of logarithms for performing complicate calculations. EXAMPLE 1 90 Chapter 2: SOLUTION CHECK The definition and three properties of logarithms are summarized in this bo. DEFINITION AND PROPERTIES: Base- Logarithms Definition log if and onl if 1 0 Verball: log is the eponent in the power of that gives Properties log log log Verball: The log of a product equals the sum of the logs of the factors. log log log Verball: The log of a quotient equals the log of the numerator minus the log of the denominator. log log Verball: The log of a power equals the eponent times the log of the base. The reason for the name logarithm is historical. Before there were calculators, base- logarithms, calculated approimatel using infinite series, were recorded could then be calculated b adding their logarithms (eponents) column-wise in one step rather than b tediousl multipling several pairs of numbers. credited with inventing this logical wa to do arithmetic that ou will eplore in logarithm actuall comes from the Greek words logos, which here means ratio, and arithmos, which means number. The most important thing to remember about logarithms is this: A logarithm is an eponent. Find if log Verif our solution numericall. B definition, the logarithm is the eponent of. So B calculator. Do not round. log which checks. 90 Chapter 2: Properties of Elementar Functions

31 EXAMPLE 2 SOLUTION CHECK EXAMPLE 3 SOLUTION EXAMPLE 4 SOLUTION Find if Verif our solution numericall. B definition,, the eponent of, is the logarithm of log B calculator. Do not round. which checks. logarithms. Show numericall that log(7 9) log 7 with the definition of logarithm. log(7 9) log log 7 log Calculate without rounding. log(7 9) log 7 log 9 This equalit agrees with the definition because (7 9) log(7 9) Add the eponents. Keep the same base. The logarithm is the eponent of. Show numericall that log 1 17 log 1 with the definition of logarithm. l o g log log 1 log Calculate without rounding. log 17 1 log 1 log 17 This equalit agrees with the definition because Subtract the eponents. Keep the same base. log 17 1 The logarithm is the eponent of. Technolog Notes CAS Activit 2-4a: Dilations of Eponential Functions in the Instructor s Resource Book has students use a CAS to eplore how horizontall translated eponential functions relate to dilations of eponential functions. Allow 20 2 minutes. Additional Class Eamples 1. Find log 37. Check our answer b showing that 37 for the value of ou found. Solution log Press log (37) on our calculator , which checks Find answer without round-off. 2. Find Check our answer b showing the log for the value of ou found. Solution Use our calculator. log Take log (answer) without round-off. The logarithm is the same as the eponent of. Section 2-4: 91 Section 2-4: Properties of Logarithms 91

32 CAS Suggestions Consider having students use a CAS to discover the properties of logarithms instead of telling the students what the properties are. The figure below shows some sample calculations a student might use to eplore the addition propert. The difference propert is a bit trickier for consistent results for students using a TI-Nspire because the programming sometimes produces an unepected result (see the last line of the figure below). Additionall, some students are surprised when the log of a power propert shows up unepectedl while the are computing data for the log of a product propert. Lead them to recognize that the addends could be thought of as the result of the log of a power propert to ease the cognitive dissonance. Alternativel, ask students to verif their understanding of logarithmic properties b using Boolean operators. These suggestions provide motivation for Problems 4 and 46 in the Problem Set. EXAMPLE SOLUTION EXAMPLE 6 SOLUTION EXAMPLE 7 SOLUTION CHECK 92 Chapter 2: Show numericall that log 3 from the log of a product propert. log 3 log log Calculate without rounding. log 3 3 log Combine like terms. This equalit derives from the product of a log propert because log 3 log( ) log log log 3 log The log of a product equals the sum of the logs of the factors. Combine like terms. of two numbers equals the sum of the logarithms of the factors. log log. Proof c log and let d log. Then c and d. B the definition of logarithm. 1 0 c d Multipl times. cd Add the eponents. Keep the same base. log c d The logarithm is the eponent of. log log log, q.e.d Substitute for c and d. logarithms algebraicall. epressions that contain logarithms. Use the properties of logarithms to find the number that goes in the blank: log 3 log 7 log log?. Check our answer numericall. log 3 log 7 log log 3 7 log 3 log 7 log B calculator log 3 log 7 log 92 Chapter 2: Properties of Elementar Functions

33 Problem Set 2-4 Reading Analsis From what ou have read in this section, what statement A logarithm is an eponent and support it with eamples. min Quick Review Q1. In the epression 7, the number 7 is called the?. Q2. In the epression 7, the number is called the?. Q3. The entire epression 7 is called a(n)?. Q4. Write 7 as a single eponential epression. Q. Write 7 as a single eponential epression. Q6. Write the epression ( ) 7 without parentheses. Q7. For the epression () 7, ou? the eponent 7 to get 7 7. Q8. 2. Q9. 9 1/2. Q. The function is called a(n) A. B. C. D. E. Inverse of a power function 1. What does it mean to sa that equals log 0.7? 2. What does it mean to sa that equals log 8? 3. B the definition of logarithm, if a log b, then 1 0??. 4. B the definition of logarithm, if 1 0 a b, then? log?. Q1. Base Q2. Eponent Q3. Eponential epression Q4. 12 Q write the value of. Then confirm that our solution is correct b raising to the given power, taking the logarithm of the result, and showing that the final result agrees with our answer.. log 6. log log log 23.8 to write as a logarithm. Then evaluate the logarithm b calculator and show that raising to that power gives a result that agrees with the given equalit calculator. Then show numericall that raising to that power, gives the argument of the logarithm. 13. log log log show that the logarithm of the answer is equal to the original eponent of log 20 log logarithms does this equalit illustrate? What propert of eponentiation does this propert come from? 22. log 120 log 30 logarithms does this equalit illustrate? What propert of eponentiation does this propert come from? 23. Find log 3, log 7, and log. Show that log log 3 log 7. What propert of logarithms does this equalit illustrate? What propert of eponentiation does this propert come from? Section 2-4: Q6. 3 Q7. Distribute Q Q9. 9 Q. B 93 PROBLEM NOTES Problems 1 44 are like the eamples and give students practice appling the definition and properties of logarithms a b 4. a log b. 1.74; , log ; , log ; , log ; , log log ; log ; log ; log ; ; ; ; ; ; log ; log ; log( ) ; log log( 4) log log 1 log 4; log log 1 log ; b c b d b c1d 22. log(30 4) log log 30 1 log 4; log log 1 log ; b c b d b c1d 23. log(3 7) log log 3 2 log 7; log c log 2 log ; b b b c2d d Section 2-4: Properties of Logarithms 93

34 Problem Notes (continued) 28. log log 7 1 log 8; log log 30 2 log ; log log 2 2 log 8; log (0.30 ) log 2; (0.30 ) ( 0.30 ) log ( ) 3 log ; ( ) ( ) log log 7; log log 00; Problems 4 and 46 ask students to prove some of the properties of logarithms. 4. Let c log, so c. Then n ( c ) n cn, so log n cn nc n log. 46. Let c log and d log, so c and d. Then c c2d, d so log _ c 2 d log 2 log. Problem 47 gives students an opportunit to see how logarithms were useful in doing long multiplication before calculators. See page 983 for answers to Problems and the CAS Problem. 24. Find log 96, log 6, and log 16. Show that log 16 log 96 log 6. What propert of logarithms does this equalit illustrate? What propert of eponentiation does this propert come from? 2. Find log 32 and log 2. Show that log 32 log 2. What propert of logarithms does this equalit illustrate? What propert of eponentiation does this propert come from? 26. this equalit illustrate? What propert of eponentiation does this propert come from? properties of logarithms. Then eplain how each result agrees with the definition of logarithm. 27. log( ) log 0.3 log log(7 8) log 7 log log(30 ) log 30 log 30. log 8 2 log 2 log log 2 log log 3 3 log 33. log 1 log log 1 log log 7 log 3 log? 36. log log 8 log? 37. log 12 log? 38. log 20 log? 39. log 8 log log 3 log? 40. log 2000 log 2 log? log 2 log? 42. log 3 log? 43. log 12? log 44.? log 2 4. Logarithm of a Power Propert Proof Problem: n n log. 94 Chapter 2: 47a , , ,683,072 47b c ; ,700,000, which agrees (to four significant digits) with the answer from part a. 46. Logarithm of a Quotient Propert Proof Problem: _ log log. 47. The Name Logarithm Problem: a. Before there were calculators, if ou had to multipl , ou would have to use long multiplication three times to then that answer b 92. Simulate this process on our calculator b multipling and writing the result, then multipling that multipling that answer b 92 and writing the final result. b. Before there were calculators, if ou had to add , ou could write the numbers in a column and add without writing down an intermediate results. Do this addition column-wise, without using a calculator: c. Base- logarithms were invented so that strings of numbers could be multiplied b adding their logarithms column-wise. You would look up the logarithms in tables, add these column-wise, and then use the tables backward to find the answer. The computation would look something like this: log 27 log log Add these logarithms column-wise, without using a calculator. Simulate finding the product b raising to the eponent ou found from adding the logarithms and rounding to four significant digits. Does the result agree with our result in part a? Additional CAS Problem Investigate the log of a product, log of a quotient, and log of a power properties using different number bases in our logarithms. Which rules, if an, also appl for logarithms with bases other than? Can ou state generic product, quotient, and power properties for logarithms? 94 Chapter 2: Properties of Elementar Functions

35 2- Objective Logarithms: Equations and Other Bases In the previous section ou learned that a logarithm is an eponent of. In this section ou will learn that it is possible to find logarithms using other positive numbers as the base. Then ou will learn how to use the properties of logarithms to solve an equation for an unknown eponent or to solve an equation involving logarithms. Use logarithms with base or other bases to solve eponential or logarithmic equations. Logarithms with An Base: The Change-of-Base Propert If 1 0, then is the base- logarithm of. Similarl, if 2, then is the base-2 logarithm of. The onl difference is the number that is the base. To distinguish among logarithms with different bases, the base is written as a subscript after the abbreviation log. For instance, 3 lo g lo g lo g The smbol lo g 2 8 is pronounced log to the base 2 of 8. The smbol lo g 0 is, of course, equivalent to log 0, as defined in the previous section. Note that in all cases a logarithm is an eponent. DEFINITION: Logarithm with An Base Algebraicall: lo g b if and onl if b, where b > 0, b 1, and > 0 Verball: lo g b means that is the eponent of b that gives as the answer. The wa ou pronounce the smbol for logarithm gives ou a wa to remember the Section 2- Class Time 1 da PLANNING Homework Assignment RA, Q1 Q, Problems 1, 2, 3 49 odd Teaching Resources Supplementar Problems Technolog Resources CAS Activit 2-a: Dilations of Logarithmic Functions TEACHING Important Terms and Concepts Common logarithm Natural logarithm e Change-of-base propert Eponential equation Logarithmic equation Section 2-: 9 Section 2-: Logarithms: Equations and Other Bases 9

36 Section Notes This section discusses logarithms log b a, where the base b must satisf the conditions b. 0 and b 1. Emphasize to students that the number log b a (pronounced log base b of a ) is the eponent to which ou raise b in order to get a, a statement that would not make sense if b did not satisf b. 0 and b 1. Eamples 1 and 2 are direct applications of the definition of a logarithm with an base. Eamples 8 on pages 98 0 demonstrate how to use the logarithmic properties to solve logarithmic equations. Eamples 6 and 8 on pages 98 0 emphasize the importance of checking the solutions ou find in the original logarithmic equation. Emphasize to students that the work the do in solving a logarithmic (or other tpe of) equation onl gives them candidates for solutions and that the need to check these solutions in the original equation. There are two built-in logarithm bases on a grapher, base and base e. Base- logarithms are called common logarithms, and base-e logarithms are called natural logarithms. Natural logarithms are ver important in calculus, so it is necessar for students to become familiar with this new base. The natural logarithm is denoted ln and pronounced el, en. To calculate logarithms with bases other than or e, students will need to use the change-ofbase propert. Eample 3 shows how to calculate log 17. Eample 4 on page 97 is another application of the change-ofbase propert. EXAMPLE 1 96 Chapter 2: SOLUTION EXAMPLE 2 SOLUTION Nautilus shells have a logarithmic spiral pattern. Write lo g c a in eponential form. Think this: is read log base, so is the base. eponent. Because the log equals a, a must be the eponent. a is the argument of the logarithm, c. Write onl this: a c Write z m in logarithmic form. lo g z m Two bases of logarithms are used frequentl enough to have their own ke on most calculators. One is base- logarithms, or common logarithms, as ou saw in the previous section. The other is base-e logarithms, called natural logarithms, where e , a naturall occurring number (like ) that ou will find advantageous later in our mathematical studies. The smbol ln (pronounced el en of ) is used for the natural logarithms: ln lo g e. DEFINITION: Common Logarithm and Natural Logarithm Common: The smbol log means lo g. Natural: The smbol ln means lo g e, where e is a constant equal to To find the value of a base-e logarithm, just press the ke on our grapher. For instance, ln 30 To show what this answer means, raise e e 30 z 4 = m log z m = 4 Use the e log c = a a = c our calculator. 96 Chapter 2: Properties of Elementar Functions

37 EXAMPLE 3 SOLUTION CHECK EXAMPLE 4 SOLUTION base Find lo g 17. Check our answer b an appropriate numerical method. = log 17 lo g eponent Use the definition of logarithm. the answer lo g lo g 17 Take lo g of both sides. lo g lo g 17 Use the log of a power propert to peel off the eponent. lo g Divide both sides b the coefficient of. lo g lo g Substitute for Do not round the to the base- logarithm of that number. The conclusion of the eample can be written this wa: lo g 17 1 lo g l og To find the base- logarithm of an number, simpl multipl its base- logarithm ). This proportional relationship is called the change-of-base propert. From the lo g 17 lo g 17 lo g Notice that the logarithm with the desired base is b itself on the left side of the equation and that the two logarithms on the right side have the same base, presumabl one available on our calculator. The bo shows this propert for bases a and b and argument. PROPERTY: The Change-of-Base Propert of Logarithms lo g a log b lo g b a or lo g a 1 lo g b a (lo g ) b Find ln 29 using the change-of-base propert with base- logarithms. Check our answer directl b pressing ln 29 on our calculator. log 29 ln 29 log e e). Directl: ln which agrees with the answer ou got using the change-of-base propert. Section 2-: Logarithms: Equations and Other Bases 97 One of the main uses of logarithms is in solving eponential equations equations in which the unknown is an eponent. (In Section 2-2, logarithms were used in this wa to find the particular equation for a power function.) When solving an eponential equation, the base of the logarithm used does not affect the answer. However, sometimes choosing a logarithm with a particular base can make solving the equation easier. Show students that the same solution is obtained whether is solved using common logarithms or natural logarithms. Then show them that solving 00 0 e is easier if natural logarithms are used, because ln e 1. Differentiating Instruction Note that the concept of log b a being a number is difficult for man students. Have students write the definition of a logarithm in their journals, both smbolicall and verball. For ELL students, perhaps in their primar language. Make sure ELL students understand what b. 0 and b 1 mean. Give eamples of bases that fit these restrictions, such as 2 and 1_ 2. Because some countries use onl log e to represent natural logarithms, clarif for ELL students that the terms ln, natural log, and log e mean the same thing and are interchangeable. Clarif that log a ( log ) ( log a) ln ln a. Section 2-: Logarithms: Equations and Other Bases 97

38 Technolog Notes CAS Activit 2-a: Dilations of Logarithmic Functions in the Instructor s Resource Book has students use a CAS to eplore how horizontall translated logarithmic functions relate to dilations of logarithmic functions. Allow 20 2 minutes. Additional Class Eamples 1. Find the missing value. ln 72 2 ln 8 ln? Solution Estimate log Find log 2 17 using the properties ou have learned in this section. Solution log because and 16 is close to 17. Use our calculator to get log 17 log 2 17 log Solve log 2 ( 2 2) 1 log 2 ( 1 3) 4. Solution log 2 ( 2 2) 1 log 2 ( 1 3) 4 log 2 ( 2 2)( 1 3) 4 ( 2 2)( 1 3) or Check both solutions in the original equation to see that is etraneous and that the onl solution is The spiral arms of galaies follow a logarithmic pattern. EXAMPLE 98 Chapter 2: SOLUTION EXAMPLE 6 SOLUTION The properties of base- logarithms presented in the previous section are generalized here for an base. Properties of Logarithms The Logarithm of a Power: lo g b lo g b Verball: The logarithm of a power equals the product of the eponent and the logarithm of the base The Logarithm of a Product: lo g b () lo g b log b Verball: The logarithm of a product equals the sum of the logarithms of the factors. The Logarithm of a Quotient: lo g b log b lo g b Verball: The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator. Solving Eponential and Logarithmic Equations how ou can do this. Solve the eponential equation algebraicall, using logarithms log 7 3 log log 7 log 983 log log Solve the equation lo g 2 ( 1) lo g 2 ( 3) 3 log 2 ( 1) lo g 2 ( 3) 3 lo g 2 [( 1)( 3)] 3 Take the base- logarithm of both sides. Use the logarithm of a power propert. Divide both sides b the coefficient of. Appl the logarithm of a product propert. 2 3 ( 1)( 3) Use the definition of logarithm. 98 Chapter 2: Properties of Elementar Functions

39 EXAMPLE 7 SOLUTION ( )( 1) 0 Solve b factoring. Reduce one side to zero. Use the smmetric propert of equalit. or 1 Solutions of the quadratic equation. You need to be cautious here because the solutions in the final step are the solutions of the quadratic equation, and ou must make sure the are also solutions of the original logarithmic equation. Check b substituting our solutions into the original equation. If, then If 1, then lo g 2 ( 1) lo g 2 ( 3) lo g 2 (1 1) lo g 2 (1 3) lo g 2 lo g is a solution, but 1 is not. Solve the equation and check our solutions. e 2 3 e 2 0 e 2 3 e 2 0 lo g 2 (2) lo g 2 ( which is undefined. CAS Suggestions Consider having students eplore the properties the learned about base- logarithms using logarithms with different bases before presenting them to the class. The TI-Nspire CAS operates as easil with bases e and as it does with an other positive number as its base. Note, however, that if a decimal value is entered for either the base or object of the logarithm, the TI-Nspire will default to a non-cas numerical approimation for the epression unless the calculator is set in eact mode. CHECK EXAMPLE 8 SOLUTION ( e ) 2 3 e 2 0 Appl the properties of eponents. You can realize that this is a quadratic equation in the variable e. Using the quadratic formula, ou get e e 2 or e 1 You now have to solve these two equations. e 2 e 1 ln e 2 ln 2 3 e ln 2 2 e e 0 2 e ln e ln (1) (2) 2 0 Both solutions are correct. Solve the logarithmic equation ln( 3) ln( ) 0 and check our solution(s). l n ( 3) ln( ) 0 ln[( 3)( )] 0 Use the logarithm of a product propert. Section 2-: 99 Section 2-: Logarithms: Equations and Other Bases 99

40 PROBLEM NOTES Supplementar Problems for this section are available at keonline. The problems in this section are trivial for students using a CAS. For an added challenge have students use a CAS to confirm their algebraic steps. This also provides them with feedback on their problem-solving techniques. Note that a CAS ma use a different notational format when making calculations. Students need to be able to adapt to the challenge of unepected formats. Understanding algebraic equivalence is a ke aspect of using a CAS effectivel. Q1. 40 Q2. 9 Q3. Q4. 3 Q Q6. Logos, arithmos Q7. Eponent Q8. 8 Q9. 40 Q. E Problems 1 34 give students practice appling the definition and properties of logarithms. 1. log b if and onl if b for. 0, b. 0, b 1 2. log a log b for. 0, a. 0, a 1, log b a b. 0, b c p 4. v 6. log k lo g m 13 d _ log Chapter 2: CHECK Problem Set 2- Reading Analsis ( 3)( ) e 0 1 Definition of (natural) logarithm or B the quadratic formula : ln(2.87 3) ln(2.87 ) which checks. ln( 3) ln( ) ln( ln( which is undefined. The onl valid solution is From what ou have read in this section, what do ou consider to be the main idea? What two bases of logarithms are found on most calculators? How do the differ? How do ou find a logarithm with a base other than these? How do ou use the logarithm of a power propert to solve an equation that has an unknown eponent? min Quick Review Q1. log log 8 log? Q2. log 36 log? Q3. log 2 2 log? Q4. log 7 3? log 7 Q. 1. log 13 log? Q6. The name logarithm comes from the words? and?. 31. log (or lo g ) _ ln ln Q7. A logarithm is a(n)?. Q8. log 6 log? Q9. log 0. log 20 log? Substitute without rounding. No logarithms of negative numbers. Q. log(2) is A. log(1/2) B. 1 log 2 C. log 2 D. log(1/2) E. Undefined 1. Write the definition of base-b logarithms. 2. State the change-of-base propert. 3. Write in eponential form: l og 7 p c 4. Write in eponential form: lo g v 6. Write in logarithmic form: k 9 6. Write in logarithmic form: m d 13 0 Chapter 2: Properties of Elementar Functions

41 Check our answer b raising the appropriate number to the appropriate power. 7. log log l og log log l og log log ln 8 ln 7 ln? 16. ln ln 20 ln? 17. ln 3? ln ln? ln ln 36 ln? ln ln? ln 7 ln ln? ln 22. ln? ln 23. ln 1? 24. ln e? 2. log? 26. log 1? 27. lo g 7 33 lo g log? ln 3? lo g lo g lo g lo g n? log lo g n? 31. ln ln? 32. log log e? 33. lo g k k 3? 34. If lo g k 2 and lo g k, then lo g k?. and check our solution. 3. log(3 7) log( 3) lo g 2 ( 3) lo g 2 ( lo g 2 (2 1) lo g 2 ( 2) ln( 9 ) ln( 2) ln( 2) Problems 3 48 require students to solve logarithmic and eponential equations , 2 24 The equation is undefined for e 46. e e 2 e Compound Interest Problem: If ou invest $,000 in a savings account that pas interest at the rate amount M in the account after ears is given b the eponential function M, a. Make a table of values of M for each ear from 0 to 6. b. How can ou conclude that the values in the c. Suppose that ou want to cash in the savings account when the amount M reaches $27,000. Set M equal to 27,000 and solve the resulting eponential equation algebraicall using logarithms. Convert the solution to months, and round appropriatel to find how man whole months must elapse before M first eceeds $27, Population of the United States Problem: Based on the 1990 and 2000 U.S. censuses, the population that time period. That is, the population at the end of an one ear was population at the beginning of that ear. a. How do ou tell that the population function b. The population in 1990 was about epressing population, P, as a function of the n ears that have elapsed since c. Assume that the population continues to grow the ear in which the population first reached 300 million. In finding the real-world answer, use the fact that the 1990 census was taken as of April 1. How does our prediction compare with that of the U.S. Census Bureau, which placed the date as October 2006? Section 2-: The equation is undefined for no solution Problems 47 and 48 can be solved on a CAS without using the Solve command b completing the square. Notice that the CAS automaticall rewrites ln(1/2) as 2ln Problems 49 and 0 are real-world problems that can be solved using logarithms. 49a. M 0,000 1, , , ,8 14, ,007 49b. Whenever ou add 1 to, ou multipl M b c. 177 mo, or 14 r 9 mo 0a. Ever time ou add one ear, the population is multiplied b b. P(n) n, with n in ears and P(n) in millions of people. 0c. 1 r 79 d; around June 19, 200; This prediction is earlier than the actual date identified b the U.S. Census Bureau. Additional CAS Problems 1. Use the E p a n d command on the epression log(2). Use the properties of logarithms to eplain the result. What do all of the terms of the result have in common? Wh? 2. Solve log B (A) log 1 () for. State an B restrictions on A and B. Wh does this propert suggest that it is appropriate to restrict the bases of logarithms to values greater than 1? See page 983 for answers to CAS Problems 1 and 2. Section 2-: Logarithms: Equations and Other Bases 1

42 Section 2-6 Class Time 1 da PLANNING Homework Assignment RA, Q1 Q, Problems 1 13 odd, 14 Teaching Resources Supplementar Problems 2-6 Objective Logarithmic Functions You have alread learned about identifing properties of several tpes of functions. In this section ou ll learn that logarithmic functions have the multipl add propert. Show that logarithmic functions have the multipl-add propert, and find particular equations algebraicall. TEACHING Important Terms and Concepts Logarithmic function Multipl add propert Section Notes This section focuses on the graphs and numerical patterns associated with logarithmic functions. Emphasize that a logarithmic function is the inverse of an eponential function with the same base. In an eponential function, the base must be greater than zero and not equal to 1, and the same is true of a logarithmic function. The range of an eponential function is. 0, so the domain of a logarithmic function is. 0. This makes it eas to eplain that one cannot take the log of 0 or a negative number. An eponential function has the add multipl propert, so a logarithmic function has the multipl add propert. Discuss the graphs of the natural and common logarithmic functions shown in Figures 2-6a and 2-6b. Emphasize that both of the functions have the same domain,. 0. Moreover, the graphs of both functions are increasing, concave down, asmptotic to the -ais, and contain the point (1, 0). It is important that students learn to visualize these graphs so that the can sketch translations and dilations of these parent functions. 2 Chapter 2: Logarithmic Functions Figures 2-6a and 2-6b show the natural logarithmic function ln and the common logarithmic function log (solid graphs). These functions are inverses of the corresponding eponential functions (dashed graphs), as shown b the fact that the graphs are reflections of the graphs of e and across the line. Both logarithmic graphs are concave down. Notice also that the -values are increasing at a decreasing rate as increases. In both cases the -ais is a vertical asmptote for the logarithmic graph. In addition, ou can tell that the domain of these basic logarithmic functions is the set of positive real numbers e ln Natural logarithm: ln The figure to the right shows the effect of varing the constant a in the natural log function a 1 ln. As ou might epect, the graphs are vertical translations of ln. Note that while ln has the epected -intercept of 1, the vertical translations give the other graphs different -intercepts. Because there was no horizontal translation, the vertical asmptote is still at the -ais. log Common logarithm: log Figure 2-6a Figure 2-6b The general equation of a logarithmic function on most graphers has constants to allow for vertical translation and dilation. = 2 + ln = ln = 2 + ln 2 Chapter 2: Properties of Elementar Functions

43 DEFINITION: Logarithmic Functions General equation: a b log c Section 2-6: Base-c logarithmic function where a, b, and c are constants, with b 0, c 0, and c 1. The domain is all positive real numbers. Transformed function: a b lo g c ( d) where a is the vertical translation, b is the vertical dilation, and d is the horizontal translation. Note: Remember that log stands for the base- logarithm and ln stands for the base-e logarithm. Multipl Add Propert of Logarithmic Functions Th e s e - and -values have the multipl add propert. Multipling b 3 results in adding 1 to the corresponding -value B interchanging the variables, ou can notice that is an eponential function of. You can find its particular equation b algebraic calculations. 2 3 This equation can be solved for as a function of with the help of logarithms. ln ln ln ln 2 ln 3 Take the natural logarithm (ln) of both sides. Use the product and power properties of logarithms. ln 3 ln ln 2 Isolate the term with. 1 ln 3 ln ln 2 ln 3 Solve for ln Calculate constants b calculator. This equation is a logarithmic function with a and b of a logarithmic function. Reversing the steps lets ou conclude that logarithmic functions have this propert in general. The figure to the right shows the effect of varing the constant d for the natural log function ln( 2 d). As ou might epect, the graphs are horizontal translations of ln. Note that while ln is asmptotic to the -ais, the horizontal translations give the other graphs different vertical asmptotes. = ln( + 2) = ln( 2) = ln 3 The figure below shows the effect of varing the multiplicative constant b in b ln. The -intercept remains at 1, and the vertical asmptote remains at the -ais. Note that if b is negative, the graph is reflected across the -ais. = 2 ln = ln = 2 ln The figure below shows that changing the base of the logarithmic function has the same effect as a vertical dilation. It is for this reason that it is not necessar to use different bases in order to fit logarithmic functions to various data sets. = log 1. = log 0. = ln The table on page 3 illustrates the multipl add propert of logarithmic functions. The particular equation for the function is found b first writing as an eponential function of and then using logarithms to write in terms of. (Part b of Eample 1 on page 4 shows a more direct method for finding a particular logarithmic function that fits a set of points.) Section 2-6: Logarithmic Functions 3

44 Section Notes (continued) Eample 1 demonstrates how to use the multipl add propert to find missing values and how to find the particular equation of a logarithmic function algebraicall. Eample 2 involves graphing three logarithmic functions and finding their domains. In parts a and b, have students sketch the graphs b appling transformations to the parent functions and then check their answers with their graphers. The function in part c is more difficult to sketch b hand, so students will want to use their graphers. Because the function involves base-2 logarithms, it must be rewritten in terms of base logarithms (or base-e logarithms) before it can be graphed. (Refer to the Additional Class Eample.) Finding the domain of each function in Eample 2 involves finding the values where the argument of the logarithmic function is greater than zero. For part c, this involves solving a quadratic inequalit. You ma need to remind students how to solve such inequalities. The solution in the tet uses the graph of the equation to find out where the quadratic function has positive values (in other words, where the graph of the parabola is above the -ais). An alternate solution to the problem is to factor into ( 2 1)( 1 1). Then the domain of f is when ( 2 1)( 1 1). 0 or when both factors have the same sign. Differentiating Instruction Have ELL students write the Multipl Add Propert in their journals and note that this is the inverse of the Add Multipl Propert of Eponential Functions. Also have them include a sketch of log 2 and 2 (similar to Figure 2-6a) and a labeled sketch of a transformed logarithmic function, so that the have a visual reference. EXAMPLE 1 4 Chapter 2: SOLUTION PROPERTY: Multipl Add Propert of Logarithmic Functions If f is a logarithmic function, then multipling b a constant results in adding a constant to the value of f (). That is, for f () a b log c, if 2 k 1, then f 2 b lo g c k f 1 Particular Equations of Logarithmic Functions You can find the particular equation of a logarithmic function algebraicall b substituting two points that are on the graph of the function and evaluating the Suppose that f is a logarithmic function with values f (3) 7 and f (6). a. Without finding the particular equation, find f (12) and f b. Find the particular equation algebraicall using natural logarithms. c. Confirm that our equation gives the value of f a Have ELL students write a summar of the section, perhaps in pairs, in place of answering the Reading Analsis questions. ELL students ma need help understanding the language in Problems 3 and 4. However, being introduced to concepts such as Carbon-14 dating and the Richter scale ma be useful in their general education f (12) 13 and f 16 b. f () a b ln Write the general equation. 7 a b ln 3 a b ln 6 Substitute the given points. 3 b ln 6 b ln 3 Subtract the equations to eliminate a. b 3 Factor out b and then divide b ln 6 ln 3 ln 6 ln 3 7 a b. a 7 f () c. B calculator, f 16, which checks. Store a and b without rounding. Write the particular equation. Graph it on our grapher. If ou assign Problem 14, let ELL students write in their primar language if the wish. 4 Chapter 2: Properties of Elementar Functions

45 EXAMPLE 2 SOLUTION f() Figure 2-6c f() Figure 2-6d f() Figure 2-6e a. f () 3 log( 1) b. f () ln( 3) c. f () lo g 2 ( 2 1) a. You can get the graph of the function f () 3 log( 1) through transformations of the parent logarithmic function: a horizontal translation b 1 unit and a vertical dilation b a factor of 3. Figure 2-6c shows the resulting graph. You know that the domain of a logarithmic function is positive real numbers, so the argument of a logarithmic function has to be positive. 1 0 The domain of the function is 1. Section 2-6: Add 1 to both sides of the inequalit. b. Figure 2-6d shows the graph of the function f () ln( 3). You can get this graph b reflecting the graph of the function ln across the -ais and translating it horizontall b 3 units. Domain: 3 0 The domain of the function is 3. Argument of a logarithmic function is positive. c. In order to graph this function on our grapher, use the change-of-base propert. f () lo g log 2 1 log 2 Figure 2-6e shows the resulting graph. Domain: Argument of a logarithmic function is positive. You can solve this inequalit graphicall. Graph the quadratic function and look for those values of for which the function value is greater than zero or the graph is above the -ais (see Figure 2-6f). The domain is 1 or Figure 2-6f Additional Class Eample Plot f () and g () log on the same screen. Use equal scales on both aes. Show that the graphs are reflections of each other across the line. Solution log log log log Use the change-of-base propert. Enter f() and g() log. The graphs are reflections of each other across the line, as shown in the figure. f() = g() = log 1 1 = CAS Suggestions Proving the general case of the multipl add propert of logarithms is straightforward on a CAS. The figure below shows that if the -coordinate in f () a 1 b log c () is multiplied b a positive constant k, the difference in the resulting -coordinates is constant. Students need to be aware of potential domain difficulties in order to obtain the desired result with a CAS. In this case, k needs to be restricted in order to give the desired result. Students often epand log(ab) to log(a) 1 log(b) without considering the signs of A and B; using a CAS encourages this awareness. Section 2-6: Logarithmic Functions

46 PROBLEM NOTES Supplementar problems for this section are available at keonline. Q1. Linear Q2. Eponential Q3. Inverse power Q4. Quadratic Q. Answers will var. Q6. P Q7. Parabola Q Q9. 48, 96, 192 Q. Eponential Problems 1 4 require students to find particular equations for logarithmic functions that fit given data sets. The then use the particular equations as mathematical models to make predictions of for given values of, and vice versa. Problems 1 and 2 can be solved with a sstem solution on a CAS. Once the function is defined, function notation can be used to show that the remaining values hold. Here ou can see the steps for Problem 1b and c. Consider demonstrating this approach before asking students to use it. t Problem Set 2-6 Reading Analsis From what ou have read in this section, what do ou consider to be the main idea? What is the general equation of a logarithmic function, and how is a logarithmic function related to an eponential function? What numerical pattern do regularl spaced values of a logarithmic function follow? min Quick Review Q1. Name the kind of function graphed in Figure 2-6g. Figure 2-6g 6 Chapter 2: 14.4 Figure 2-6h Q2. Name the kind of function graphed in Figure 2-6h. Q3. Name the kind of function graphed in Figure 2-6i. Figure 2-6i Figure 2-6j Q4. Name the kind of function graphed in Figure 2-6j. Q. Name a real-world situation that could be modeled b the function in Figure 2-6k. Figure 2-6k 1a b ln 3.6 ln ln 26 ln ln 1c. The equation fits the data. Q6. Sketch a reasonable graph: The population of a cit depends on time. Q7. The graph of a quadratic function is called a?. Q8. 7 ) 2 Q9. Write the net three terms in this sequence: Q.? functions. a. Show that the values in the table have the b. Use the first and last points to find algebraicall the particular equation of the natural logarithmic function that fits the points. c. Verif that the equation in part b gives the other points in the table a b ln ln ln 2c. The equation fits the data. 6 Chapter 2: Properties of Elementar Functions

47 3. Carbon-14 Dating Problem: The ages of things, such as wood, bone, and cloth, that are made from materials that were once living can be determined b measuring the percentage in them. This table contains data on age as a function of the remaining percentage of Percentage Remaining Age (r) c. You can use our mathematical model to interpolate between the given data points to find fairl precise ages. Suppose that a piece content 73.9%. What would ou predict its age to be? d. How old would ou predict a piece of wood to e. Search on the Internet or in some other resource to find out about earl hominid Give the source of our information. 4. Earthquake Problem: You can gauge the amount of energ released b an earthquake b its Richter magnitude, a scale devised b seismologist Charles F. Richter in 193. The Richter magnitude is a base- logarithmic function of the energ released b the earthquake. These data show the Richter magnitude m for earthquakes that release energ equivalent to the eplosion of tons of TNT (tri-nitro-toluene). (tons) m (Richter magnitude) 1,000 1,000, Problems 3 allow students to focus on the question while the CAS does the algebra. 3a. The inverse of an eponential function is a logarithmic function. 3b. 238, , ln p; r , ver close to 21 3c. (73.9) ears old 3d. (20) 13, ,300 ears old 3e. Answers will var. 4a. m log The skull of the saber-toothed cat, which lived in the Pleistocene more than 11,000 ears ago. a. Based on theoretical considerations, it is remaining is an eponential function of the age. How does this fact indicate that the age should be a logarithmic function of the percentage? b. Using the first and last data points, find the particular equation of the logarithmic function that goes through the points. Show that the equation gives values of other points close to those in the table. The director of the National Earthquake Service in Golden, Colorado, studies the seismograph displa of a magnitude 7. earthquake. a. Find the particular equation of the common logarithmic function m a b log that fits the two data points. Section 2-6: 7 Section 2-6: Logarithmic Functions 7

48 Problem Notes (continued) 4b. m( 3 9 ) ; m( ) ; m c billion tons 4d. False. Doubling the energ increases the Richter magnitude linearl b 2_ 3 log points. This is not surprising, because logarithmic functions have the multipl add propert. 4e. Answers will var. Problem asks students to epress the same function both as a common logarithm function with a dilation and as a log function with a base other than. a. g () 6 lo g ; log log b. The -intercept of g is e 23, and the -intercept of h is e. Problem 6 asks students to eplore the relationship between an eponential function and its inverse. 6a. 2 6b. lo g 2 or log or ln log 2 ln 2 6c. The graph matches the dotted function. 6d. This graph also matches the dotted function. 6e. In parametric mode, graph (t) f(t), (t) t. f() : b. Use the equation to predict the Richter magnitude for strongest on record, which released the energ of billion tons of TNT An earthquake that would release the da, 160 trillion tons of TNT Blasting done at a construction site that releases the energ of about 30 lb of TNT c. The earthquake that caused a tsunami in the about 9.0. How man tons of TNT would it take to produce a shock of this magnitude? d. True or false? Doubling the energ released b an earthquake doubles the Richter magnitude. Give evidence to support our answer. e. in some other resource. Name one thing ou learned that is not mentioned in this problem. Give the source of our information.. Logarithmic Function Vertical Dilation and Translation Problem: g Figure 2-6l f g f h Figure 2-6m a. Figure 2-6l shows the graph of the common logarithmic function f () lo g (dashed) and a vertical dilation of this graph b a factor of 6, g () (solid). Write an equation of g (), considering it as a vertical dilation. Write another equation of g () in terms of l og b, where b is a number other than. Identif the base. b. Figure 2-6m shows the graph of f () ln (dashed). Two vertical translations are also shown, g () 3 ln and h() 1 ln. Find algebraicall or numericall the -intercepts of the graphs of g and h. 6. Logarithmic and Eponential Function Graphs Problem: Figure 2-6n shows the graph of an eponential function f () and its inverse function g (). 1 1 f Figure 2-6n a. The base of the eponential function is an integer. Which integer? b. Write the particular equation of the inverse function g () f 1 (). c. Confirm that our answers to parts a and b are correct b plotting on our grapher. d. With our grapher in parametric mode, plot these parametric equations: 1 (t) f (t) 1 (t) t What do ou notice about the resulting graph? e. From our answer to part d, eplain how ou could plot on our grapher the inverse of an given function. Show that our method works b plotting the inverse of the function their domains. 7. f () 2 log( 3) 8. f () log(3 2) 9. f () lo g 3 2. f () ln f () ln f () 2 (3 ) g f 21 (), given b (t) t t t 2 1, (t) t: 6 8 Chapter 2: 7. Domain: Domain: Chapter 2: Properties of Elementar Functions

49 This problem prepares ou for the net section. 13. The Definition of e Problem: Figure 2-6o shows the graph of (1 2 ) 1/. If 0, then is undefined because of division b zero. If is close to zero, then 1_ is ver large. For instance, ( ) 1/ Limit e Figure 2-6o a. Reproduce the graph in Figure 2-6o on our grapher. Use a window that has a grid point at 0. Trace to values close to zero, and record the corresponding values of. b. Two competing properties influence the epression (1 ) 1/ as approaches zero. A number greater than 1 raised to a large power is ver large, but 1 raised to an power is still 1. Which of these competing properties wins? Or is there a compromise at some number larger than 1? c. Call up the number e on our grapher. If it does not have an e ke, calculate e 1. What do ou notice about the answer to part b and the number e? 9. Domain: Research Project: On the Internet or in some other reference source, find out about contributions to the mathematics of logarithms. See if ou can find out wh natural logarithms are sometimes called Napierian logarithms. Give the source of our information Section 2-6: These rods are called Napier s bones. Invented in the earl 1600s, the made multiplication, division, and the etraction of square roots easier.. Domain:. 2 or, Problem 13 can be used as the basis for an interesting class discussion. The domain of the function is. 21 and 0. However, some graphers ma show discrete points to the left of 21. (The grapher is not familiar with our restriction that if a base is raised to a power, then the base must be positive.) The function has asmptotes at 21 and 1 and a removable discontinuit (a hole) at (0, e). Consider talking informall about the behavior of the function as approaches 21 and as approaches infinit. Another interesting aspect of this function is what s happening as approaches 0. This is a good opportunit for an informal discussion of limits. One of the definitions of e is 1/ e lim (1 1 ) 0 13b. The two properties balance out, so that as approaches 0, approaches c. e ; the are the same. Problem 14 is a research project that can be assigned for etra credit. 14. Answers will var. Additional CAS Problems 1. A given logarithmic function contains the points (3, ) and (7, 13). a. If the function is a common logarithm, determine its general form. b. If the function is a natural logarithm with a vertical asmptote at 2, determine its translated form. c. What is the base of the function if onl dilations were applied to make the function contain the given points? 2. The graph of a non-transformed eponential function intersects the graph of a non-transformed logarithmic function twice. If each function has the same base, what can be said about the value of the base of the functions? See page 983 for answers to Problems 11 13a and CAS Problems 1 2. Section 2-6: Logarithmic Functions 9

50 Section 2-7 Class Time 1 da PLANNING Homework Assignment RA, Q1 Q, Problems 1, 3, 4,, 7 Teaching Resources Eploration 2-7: The Logistic Function for Population Growth Blackline Master Eploration Problem 2 Supplementar Problems Test, Sections 2-4 to 2-7 Forms A and B Technolog Resources Presentation Sketch: Logistic Present.gsp Eploration 2-7, The Logistic Function for Population Growth TEACHING Important Terms and Concepts Logistic function Restrained growth 2-7 Logistic Functions for Restrained Growth Suppose that the population of a new subdivision is growing rapidl. This table shows monthl population figures. (mo) (houses) Figure 2-7a shows the plot of points and a smooth (dashed) curve that goes through them. You can tell that it is increasing, is concave up, and has a positive -intercept, suggesting that an eponential function fits the points. Using the first and last points gives the function ( ), the curve shown in the figure, which fits the points almost eactl. Suppose that there are onl 00 lots in the subdivision. The actual number of houses will level off, approaching 00 graduall, as shown in Figure 2-7b. 20 Concave up Figure 2-7a Restrained growth Figure 2-7b Section Notes A logistic function can be used to model restrained population growth. From the left to right, the graph of a logistic function (with b. 0) starts out looking ver much like an increasing eponential function. It is asmptotic to 0 and is increasing and concave up. At the point of inflection the rate of growth is the greatest. After the point of inflection the rate of growth slows, the function becomes concave down, approaching a horizontal asmptote. The maimum value the function approaches is called the carring capacit, or the maimum sustainable population. The -coordinate of the point of inflection is equal to one-half of c, the carring capacit. The logistic function is smmetric about the point of inflection. Objective 1 Chapter 2: A logistic function is the onl tpe of function in this chapter that changes concavit. At the point of inflection, the rate of growth of the function is a maimum. Eploration 2-7 asks students to fit a logistic growth model to some population data. In the problem the maimum sustainable population is 43,000 people, so c 43. The values of a and b are calculated b using the first and last data points. In this section ou will learn about logistic functions that are useful as mathematical models of restrained growth. In this eploration, ou ll fit the logistic function to restricted population growth. Eample 1 provides another opportunit to find a logistic model for a set of data and to eamine the role of the inflection point. When ou discuss this eample, be sure to emphasize the real-world importance of the point of inflection. The properties bo after the eample summarizes the features of logistic functions. Notice that the bo includes the case where b, 0, in which the logistic function is decreasing. Problem 6 involves 1 Chapter 2: Properties of Elementar Functions

51 E XPLORATION 2-7: The Logistic Function for Population Growth Suppose that this table lists the population of a 3. Toward the end of the -r period, the small communit, in thousands of people. The figure shows a scatter plot of the data. function seems to be leveling off. A function that models such population growth is the logistic function. Its general equation is (ears) (thousands c of people) 1 a e b 1 2 where and are the variables, e is the base of the natural logarithm, and a, b, and c stand 2 3 for constants. The communit has room 3 c 9 Calculate a and b using the first and the tenth 13 points. Write the particular equation, and (thousands of people) plot it on the same screen as the data. Sketch 40 the result What does the logistic function indicate the 8 32 (ears) population was at time 0 r? What graphical evidence do ou have that 39 the maimum population in the communit logistic in a dictionar, and 2. At first the population seems to be increasing find the origin of the word. eponentiall with time. On a cop of the given graph, sketch the graph of an 7. What did ou learn as a result of doing this eponential function that would fit the first eploration that ou did not know before? si data points reasonabl well. 1 Figure 2-7c f g Figure 2-7c shows the graphs of 2 f () 2 and g () 2 1 Function f is an eponential function, and function g is a logistic function. For large positive values of, the graph of g levels off to 1. This is because, for large values of, 2 is large compared to the 1 in the denominator. So the denominator is not much different from the 2 in the numerator, and the fraction representing g () approaches 1. g () For large negative values of, the 2 in the denominator is close to zero. So the denominator is close to 1. Thus the fraction representing g () approaches 2. 2 g () a real-world situation that can be modeled with a decreasing logistic function. Eploration Notes Eploration 2-7 demonstrates how to fit a logistic function to restricted population growth from two points and the upper horizontal asmptote. A blackline master for Problem 2 is available in the Instructor s Resource Book. Allow approimatel 1 minutes for this activit. Section 2-7: Sample equation: b a e There were approimatel 1134 people.. The graph seems to be leveling off at about 43, From Greek logos, meaning word or calculation 7. Answers will var. Differentiating Instruction Help ELL students with Problem 13 and let them answer Problem 14 in their primar language. Some languages have more than one word for grow, depending on what is growing; this can make discussion of growth harder for man ELL students to understand. Visualizations will help. Have students write the logistic equation and its properties in their journals. Visuall eplain increasing at an increasing rate and increasing at a decreasing rate. Have students put this in their journals. Have ELL students do Problem 14 in pairs. Section 2-7: Logistic Functions for Restrained Growth 111

52 Technolog Notes Presentation Sketch: Logistic Present.gsp at demonstrates the different logistic growth models as the parameters a, b, and c change. Additional pages of the document provide a brief introduction to the iterative approach to logistic growth. Eploration 2-7 asks students to fit a logistic growth model to some population data. The data can be plotted and the function can be graphed in Fathom. You might encourage students to use sliders in order to find the best a and b to fit the data and then to compare their values to those the find algebraicall, as the eploration suggests. As ou can see in Figure 2-7c, the logistic function is almost indistinguishable from the eponential function for large negative values of. But for large positive values, the logistic function levels off, as did the number of occupied houses represented b Figure 2-7b. You can fit logistic functions to data sets b using the same dilations and translations ou have used for other tpes of functions. You ll General Logistic Function You can transform the equation of function g in Figure 2-7c so that onl one eponential term appears. 2 g () Multipl b a clever form of 1. To get a general function of this form, replace the 1 in the numerator with a constant, c, to give the function a vertical dilation b a factor of c. Replace the eponential term 2 with ab or with a e b if ou want to use the natural eponential function. The result is shown in the bo. DEFINITION: Logistic Function General Equation f () c 1 a e b or f () c 1 a b where a, b, and c are constants and the domain is all real numbers EXAMPLE 1 Use the information on the occupied houses from the beginning of the section. (months) (houses) a. Given that there are 00 lots in the subdivision, use the points for 2 mo and mo to find the particular equation of the logistic function that satisfies these constraints. b. the result. 112 Chapter 2: 112 Chapter 2: Properties of Elementar Functions

53 SOLUTION c. Make a table showing that the logistic function fits all the points closel. d. Use the logistic function to predict the number of houses that will be occupied at the value of corresponding to 2 r. Which process do ou use, etrapolation or interpolation? e. Find the value of at the point of inflection. What is the real-world meaning of the fact that the graph is concave up for times before the point of inflection and concave down thereafter? Start with the second form of the logistic function. a a b The vertical dilation is ab a b Substititute points (2, 3) and (, 167) CAS Suggestions Students can use a CAS to produce logistic functions in different forms, gathering useful information about the curve from each form. 3 3a b a b 00 3a b a b a b 3a b Divide to eliminate a and simplif. b b / Store without rounding. 3a ( ) Substitute for b. a ( ) b. Figure 2-7b on page 1 shows the graph. c Start without rounding (months) (houses) Logistic Function (eact) (close) (close) 8 (close) (eact) Write the particular equation. d. Trace the function to Section 2-7: 113 Section 2-7: Logistic Functions for Restrained Growth 113

54 PROBLEM NOTES Supplementar problems for this section are available at keonline. Q1. Add multipl Q2. Multipl multipl Q3. Logarithmic Q4. Multipl add Q. e Q6. p h m Q7. j lo g c Q deg/s Q9. a 2 1 b 1 c, a 0 Q. D Problems 1 and 2 are similar to the eample in the tet. You ma want to remind students that the equation in Problem 1 can be rewritten to look like the form of the logistic function on page a. f() The process is etrapolation because given points. e. The point of inflection is halfwa between the -ais and the asmptote at 00. Trace the function to a value that is close to 00. Use the intersect feature to find occurs at about 33 mo. Before 33 mo, the number of houses is increasing at an increasing rate. After 33 mo, the number is still increasing but at a decreasing rate. Note that if a, b, and c are all positive, the logistic function will have two horizontal asmptotes, one at the -ais and one at the line c. The point of inflection occurs halfwa between these two asmptotes. PROPERTIES: Logistic Functions c The logistic function is where a, b, and c are constants such that 1 a e b a 0, b 0, c 0. The domain is all real numbers. The logistic function has 0 and another at c c_ 2 If b 0 If b 0 c 1 ae b Note: a 0, c 0 Asmptote at 0 Asmptote at c Point of inflection at c 2 Logistic function c 1 ae b Asmptote at c Asmptote at 0 Note: a 0, c 0 Point of inflection at c 2 Logistic function g() 1b. The graphs are almost the same for large negative values of, but widel different for large positive values of. 1c. 0; concave up for, 0 and concave down for. 0 1d. As grows ver large, the 1 in the denominator becomes insignificant in comparison to the 2.2, so g () e. g () 1 2. A table of values shows that the epressions are equivalent. Problem Set 2-7 Reading Analsis From what ou have read in this section, what do ou consider to be the main idea? What is the main difference between the graph of a logistic function and the graph of an eponential function? For what kind of real-world situations are logistic functions reasonable mathematical models? 114 Chapter 2: min Quick Review Q1. An eponential function has the?? propert. Q2. A power function has the?? propert. Q3. The equation 3 ln defines a? function. 114 Chapter 2: Properties of Elementar Functions

55 Q4. The function in?? propert. Q. The epression ln is a logarithm with the number? as its base. Q6. Write in eponential form: h lo g p m Q7. Write in logarithmic form: c j Q8. If an object rotates at 0 revolutions per minute, how man degrees per second is this? Q9. Write the general equation of a quadratic function. Q. The function g () 3 f 6 is a horizontal translation of function f b A. 3 B. C. D. 6 E Given the eponential function f () 2. 2 and 2. 2 the logistic function g () , a. domain. Sketch the result. b. How do the two graphs compare for large positive values of? How do the compare for large negative values of? c. Find graphicall the approimate -value of the point of inflection for function g. For what values of is the graph of function g concave up? Concave down? d. has a horizontal asmptote at 1. e. Transform the equation of the logistic function so that an eponential term appears onl once. Show numericall that the resulting equation is equivalent to g () as given. 2. Figure 2-7d shows the graph of the logistic function f () 3 e 0.2 e f() a. horizontal asmptote at 3. b. Read the point of inflection from the graph. Find the -coordinate algebraicall. c. For what values of is the graph concave up? Concave down? d. Transform the equation so that there is onl one eponential term. Confirm b graphing that the resulting equation is equivalent to f () as given. 3. Spreading the News Problem: You arrive at school and meet our mathematics teacher, who tells ou toda s test has been canceled! You and our friend spread the good news. The table shows the number of students,, who have heard the news after time, in minutes, has passed since ou and our friend heard the news. (min) (students) a. points. Is the graph of this function concave up, concave down, or both? b. There are 1220 students in the school. Use the numbers of students at time 0 min and at time function that meets these constraints. c. first 3 hours. d. Based on the logistic model, how man students have heard the news at 9:00 a.m. if ou heard it at 8:00 a.m.? How long will it be until all but ten students have heard the news? Problem 2b can be addressed using a Solve command on a CAS. 2a. As grows ver large, the 4 in the denominator becomes insignificant in comparison to e 0.2 so f() 3 e 0.2 e e 0.2 e b. Point of inflection at c. f is concave up for, and concave down for d. f() The graphs e coincide. Problems 3 and are real-world applications of logistic modeling. 3a. Concave up b (609)(1.19 ) 2 3c Figure 2-7d 30 Section 2-7: d. 43 students; min Section 2-7: Logistic Functions for Restrained Growth 11

56 Problem Notes (continued) Problem 4 is an interesting class simulation eperiment. If ou have the time, it is worthwhile. If the same number is generated a second time, ignore it and generate another random number. 4. Simulations will var. a. Concave down b (2.732 )( ) 2 c d. The point of inflection occurs at ( , 198). Before approimatel 12 das passed, the rate of new infection was increasing; after that, the rate was decreasing. e. Approimatel 363 people were infected. f. Answers will var. Problem 6 presents a real-world situation that is modeled b a decreasing logistic function. In part c, there are too man rabbits to be supported b the environment, so the begin to die off faster than the are born, thereb reaching the maimum sustainable population. Note that when the -coordinate of the initial value is greater than 00, the logistic function decreases. 4. Spreading the News Simulation Eperiment: In this eperiment ou will simulate the spread of and then selects two people at random to tell the news to. Do this b selecting two random integers between 1 and the number of students in our class, inclusive. (It is not actuall necessar to tell an news!) The random number generator on one student s calculator will help make the random selection. The two people with the chosen numbers stand. Thus, after the first iteration, there will probabl be three students standing (unless a duplicate random number two more people to tell the news to b selecting a total of si (or four) more random integers. Do this for a total of ten iterations or until the entire class is standing. At each iteration, record the number of iterations and the total number of people who have heard the news. Describe the results of the eperiment. Include things such as The plot of the data points. A function that fits the data, and a graph of chose the function ou did. A statement of how well the logistic model fits the data. The iteration number at which the good news was spreading most rapidl.. Ebola Outbreak Epidemic Problem: In the out in the Gulu district of Uganda. The table shows the total number of people infected from infections. The final number of people who were infected during this epidemic was a virus that causes internal bleeding and is fatal in most cases.) 116 Chapter 2: e 2 6a. a 9, so f() 00. The graph is correct. 6b. The natural ceiling on the number of rabbits is 00. If the population is less than this, it will grow toward this limit. 6c. a 2 1_ 00 2, so g () 1 2 1_. The graph 2 e 2 is correct. The sign of a is negative, whereas the definition of logistic function states that a 0, so this is a generalization of the definition. (das) (total infections) A Red Cross medical officer instructs villagers about the Ebola virus in Kabede Opong, Uganda. a. fits the data. Is the graph of this function concave up or concave down? b. Use the second and last data points to find the particular equation of a logistic function that fits the data. c. the logistic function from part b. Sketch the results. d. Where does the point of inflection occur in the logistic model? What is the real-world meaning of this point? e. Based on the logistic model, how man people f. Find data about other epidemics. Give our source. Tr to model the spread of the epidemic for which ou found data. 6d. If the population is greater than the number the region can support, it will decrease toward that limit. 116 Chapter 2: Properties of Elementar Functions

57 6. Rabbit Overpopulation Problem: Figure 2-7e shows two logistic functions represented b the equation 00 1 ae Both functions represent the population of rabbits in a particular woods as a function of time, in ears. The value of the constant a is to be determined under two different initial conditions g f Figure 2-7e d. How do ou interpret the mathematical model under the condition of part c? What seems to be the implication of tring to stock a region with a greater number of a particular species than the region can support? 7. Given the logistic function f () c 1 a e a. a graphs of f for c 1, 2, and 3. Use as a domain. Sketch the results. True or false? The constant c is a vertical dilation factor. b. Figure 2-7f shows the graph of f with c 2 and with a 0.2, 1, and. Which graph is which? How does the value of a transform the graph? f() 2 7b. Changing a seems to translate the graph horizontall. 2 f() a 0.2 a 1 a 7c. Horizontal translation b 3 2 f() g() 7d. a e a. For f () in Figure 2-7e, 0 rabbits were introduced into the woods at time 0. Find the value of the constant a under this condition. Show that our answer is correct b plotting the graph of f on our grapher. b. How do ou interpret this mathematical model with regard to what happens to the rabbit population under the condition in part a? c. For g () in Figure 2-7e, 2000 rabbits were introduced into the woods at time 0. Find the value of a under this condition. Show that the graph agrees with Figure 2-7e. How does the sign of a represent a generalization of the definition of logistic function? Problem 7 deals with dilations and translations of the parent logistic function and reviews material from Chapter 1. Section 2-7: Figure 2-7f c. g () c 1 a e. 3) What transformation applied to f does this represent? Confirm that our answer is correct b plotting f and g on the same screen using c 2 and a 1. d. What value of a in the equation of f () would produce the same transformation as in part c? 7a. True. 2 f() c 3 c 2 c Additional CAS Problems 1. A logistic function contains the points (21, 6), (0, 3), and (1, 1.2). Determine an equation for the function. What are the coordinates of the inflection point of the function? 2. What are the coordinates of the inflection point of a logistic function in the form c (1 1 a b )? 3. Two common forms of logistic-function equations are c (1 1 a b ) and c 1 1 b. (2d) What is the graphical effect of each parameter: a, b, c, and d? Which two parameters control horizontal translations? Under what algebraic conditions are these two equations equivalent? How does this confirm the identification of the parameters controlling horizontal translations? See page 983 for answers to CAS Problems 1 3. Section 2-7: Logistic Functions for Restrained Growth 117

58 Section 2-8 PLANNING Class Time 2 das (including 1 da for testing) Homework Assignment Da 1: R0 R7, T1 T28 Da 2: Problem Set 3-1 (after Chapter 2 Test) Teaching Resources Eploration 2-8a: Rehearsal for Chapter 2 Test Blackline Master Problem C3 Test 6, Chapter 2, Forms A and B 2-8 Chapter Review and Test In this chapter ou have learned graphical and numerical patterns for various tpes of functions: These patterns allow ou to tell which tpe of function might fit a given real-world situation. Once ou have selected a function that has appropriate the particular equation b calculating values of the constants. You can check our work b seeing whether the function fits other given points. Once ou have the correct equation, ou can use it to interpolate between given values or etrapolate beond given values to calculate when ou know, or to calculate when ou know. Section Notes TEACHING Section 2-8 contains a set of review problems, a set of concept problems, and a chapter test. The review problems include one problem for each section in the chapter. You ma wish to use the chapter test as an additional set of review problems. Encourage students to practice the nocalculator problems without a calculator so that the are prepared for the test problems for which the cannot use a calculator. Differentiating Instruction Go over the review problems in class, perhaps b having students present their solutions. You might assign students to write up their solutions before class starts. Have students write Problem C1 in their journals. Consider allowing ELL students to skip Problem C2. ELL students ma need help with the language in Problem C3. Clarif the concept of slope field in Problem C3 for ELL students. Review Problems R0. Update our journal with what ou have R1. This problem concerns these five function learned in this chapter. Include things such values: as the definitions, properties, and graphs of the functions listed. Show tpical graphs of f () the various functions, give their domains, and make connections between, for eample, the logarithmic functions. Show how ou can use logarithms and their properties to solve 30.0 for unknowns in eponential or logarithmic a. On the same screen, plot the data points and equations, and eplain how these equations the graph of f () 0.3 arise in finding the constants in the particular. equation of certain functions. Tell what ou have b. Is the function increasing or decreasing? Is the learned about the constant e and where it is used. graph concave up or concave down? c. Name the function in part a. Give an eample in the real world that this function might model. Is the -intercept of f reasonable for this real-world eample? 118 Chapter 2: Properties of Elementar Functions Because man cultures norms highl value helping peers, ELL students often help each other on tests. You can limit this tendenc b making multiple versions of the test. Consider giving a group test the da before the individual test, so that students can learn from each other as the review, and the can identif what the don t know prior to the individual test. Give a cop of the test to each group member, have them work together, and then randoml choose one paper from the group to grade. Grade the test on the spot, so students know what the need to review further. Make this test worth 1_ 3 the value of the individual test, or less. ELL students ma need more time to take the test. ELL students will benefit from having access to their bilingual dictionaries while taking the test. 118 Chapter 2: Properties of Elementar Functions

59 R2. a. Find the particular equation of a linear function containing the points (7, 9) and (, 11). Give an eample in the real world that this function could model. b. Sketch two graphs showing a decreasing eponential function and an inverse variation power function. Give two was in which the graphs are alike. Give one wa in which the are different. c. How do ou tell that the function graphed in Figure 2-8a is an eponential function, not a power function? Find the particular equation of the eponential function. Give an eample in the real world that this eponential function could model (2, 6) (, 16) Figure 2-8a d. Find the particular equation of the quadratic function graphed in Figure 2-8b. How does the equation ou find show that the graph is concave down? Give an eample in the real world that this function could model. 20 (2, 1.2) (4, 18.8) (6, 12.8) Figure 2-8b e. A quadratic function has the equation 3 2( ) 2. Where is the verte of the graph? What is the -intercept? R3. For each table of values, tell from the pattern whether the function that fits the points is linear, quadratic, eponential, or power. Eploration Notes Eploration 2-8a ma be used as an in-class rehearsal for the chapter test, or as a review assignment for homework. a. f () c. h () Section 2-8: b. g () d q () e. Suppose that f (3) 90 and f (6) 120. Find f (12) if the function is i. An eponential function ii. A power function iii. A linear function f. of eponential functions is true for the function f () b showing algebraicall that adding the constant c to multiplies the corresponding f ()-value b a constant. R4. a. The most important thing to remember about logarithms is that a logarithm is?. b. Write in logarithmic form: z p c. What does it mean to sa that log 30 d. Give numerical eamples to illustrate these logarithmic properties: i. log() log log ii. log log log iii. log log e. log log? R0. Journal entries will var. R1a. 30 f() R1b. Increasing for. 0, decreasing for, 0, concave up PROBLEM NOTES R1c. Quadratic power function. Realworld interpretations ma var. Problems R2a and R2c could be solved using sstems on a CAS. R2a. 2_ Real-world interpretations ma var. R2b. Both are decreasing and have the -ais as an asmptote. The eponential function crosses the -ais, whereas the inverse function has no -intercept (and has the -ais as an asmptote). See the graphs above. R2c. The -intercept is nonzero. ( )( ). Real-world interpretations ma var. R2d ; the coefficient of 2 is negative, which indicates the graph is concave down. Real-world interpretations ma var. R2e. Verte (, 3); -intercept: 3 R3a. Eponential R3b. Power (inverse variation) R3c. Linear R3d. Quadratic R3e. i. f (12) 213 1_ 3 R3e. ii. f (12) 160 R3e. iii. f (12) 180 R3f. f ( 1 c) c c 1.3 c f () Problems R4e and Rc Re can be solved directl on a CAS without an knowledge of the underling logarithm laws. See page 984 for answers to Problem R4. Section 2-8: Chapter Review and Test 119

60 Problem Notes (continued) Ra. Rb. Rc. 63 c p m lo g Rd. 4, the equation is undefined if 23. Re log log 3 R6a. f 1 () and f 2 () are reflections of each other across the line. f 2 () f 1 () R6b. f() ; g () 4.3 e R6c. Multipl add propert; 213 lo g 2 R6d R. a. Write in eponential form: p lo g c m b. Find lo g c. ln 7 2 ln 3 ln? d. Solve the equation: log( 1) log( 2) 1 e. Solve the equation: R6. a. On the same screen, plot the graphs of f 1 () ln and f 2 () e. Use the same scale on both aes. Sketch the results. How are the two graphs related to each other and to the line? b. For the natural eponential function f () e, write the equation in the form f () ab. For the eponential function g (), write the equation as a natural eponential function. Sunlight Under the Water Problem (R6c R6e): The intensit of sunlight underwater decreases with depth. The table shows the depth,, in feet, below the surface of the ocean ou must go to reduce the intensit of light to the given percentage,, of what it is at the surface. d. On the same screen, plot the data and the logarithmic function. Sketch the result. e. Based on this mathematical model, how deep do ou have to go for the light to be reduced to 1% of its intensit at the surface? Do ou find this b interpolation or b etrapolation? R7. a. same screen and sketch the results. f () 2 2 g () 2 b. f () is ver close to g () when when is a large positive number, f () is close to and g () is ver large. c. Transform the equation of f () in part a so that it has onl one eponential term. d. Transform the equation of g () in part a so that it is epressed in the form g () e k. e. Population Problem: A small communit is built on an island in the Gulf of Meico. The population grows steadil, as shown in the table. (months) (people) R6e ft deep (b etrapolation). R7a g() f() 1 20 R7b. When is a large negative number, the denominator of f() is essentiall equal to, so for large negative, f() g (). But for large positive, the in the denominator of f() is negligible compared with the 2 ; so f() (%) Depth (ft) c. What numerical pattern tells ou that a logarithmic function fits the data? Find the particular equation of the function. 120 Chapter 2: Properties of Elementar Functions Problem R7c appears to be surprisingl immune to the TI-Nspire CAS. The desired output formatting of the CAS prevents the user from getting a clean result when dividing the numerator and denominator b 2. You could appl the E p a n d o r Propfrac commands to the logistic function, but this approach requires some quite sophisticated transformations to obtain the desired results. Alternativel, be a reasonable mathematical model for population as a function of time. If the find the particular equation of the logistic function that contains the points for 6 mo approimatel the correct solutions for sketch the result. When is the population predicted to reach 9% of the capacit? ou could deal with the numerator and denominator individuall, but the effort involved would be far greater than simpl solving the problem without a CAS. R7c. f() (ln 2) R7d. g () e 120 Chapter 2: Properties of Elementar Functions

61 Concept Problems C1. Rise and Run Propert of Quadratic Functions C2. Log-log and Semilog Graph Paper Problem: Problem: The sum of consecutive odd counting f () be the number of numbers is alwas a perfect square. For bacteria remaining in a culture over time, in instance, g () be the area of skin, in square centimeters, on a snake of length, in centimeters. Figure 2-8d shows the graph of the eponential function f plotted on semilog graph paper. Figure 2-8e on the net page shows on the graph of the power function g plot ted on log-log graph paper. On these graphs, one This fact can be used to sketch the graph of a or both aes have scales proportional to the quadratic function b a rise-run technique logarithm of the variable s value. Thus the scales similar to that used for linear functions. are compressed so that a wide range of values Figure 2-8c shows that for 2, ou can start can fit on the same sheet of graph paper. For at the verte and use the pattern over 1, up 1; these two functions, the graphs are straight lines. over 1, up 3; over 1, up ;.... f() = C1a Figure 2-8c a. On graph paper, plot the graph of 2 b using this rise-run technique. Use integer values of pattern for values of from 0 to b. The graph of ( 2 ) 2 is a translation of the graph of 2 the verte, and then plot the graph on graph paper using the rise-run pattern. c. The graph of 0.3( 2) 2 is a vertical dilation of the graph in part b. Use the rise-run technique for this function, and then plot its graph on the same aes as in part b C1b. Verte at (2, 2) Section 2-8: R7e. The size of the population would be limited b the capacit of the island. 400 f() ( )( ) f(12) f(18) months 121 See page 984 for answers to Problem C1c. Section 2-8: Chapter Review and Test 121

62 Problem Notes (continued) Problems C2 and C3 are good problems for a research project or for etra credit. A blackline master for Problem C3 is available in the Instructor s Resource Book. C2a. f(9) ; g (60) 324 C2b. The graphs look linear. C2c. log f() log 00 1 log 0.6; -intercept is log 00; slope is log 0.6. The graph is linear. C2d. log g () log log ; -intercept is log 0.09; slope is 2 C3a. i e e 0.7 C3a. ii e e 0.7 C3a. iii. 22, e e 0.7 C3b. C3c. The graphs follow the direction of the line segments c 0.3 c 6 c g() = a. Read the values of f (9) and g (60) from the graphs. Then calculate these numbers algebraicall using the given equations. If our graphical answers are different from our calculated answers, eplain what mistakes ou made in reading the graphs. b. You ll need a sheet of semilog graph paper and a sheet of log-log graph paper for graphing. On the semilog paper, plot the function h() 2 1. using several values of in the domain [0, 1]. On the log-log paper, plot the function p() using several values of in the domain [1, 0]. What do the graphs of the functions look like? c. Take the logarithm of both sides of the equation f () Use the properties of logarithms to show that log f () is a linear function of how this fact is connected to the shape of the graph. d. Take the logarithm of both sides of the equation g () Use the properties of logarithms to show that log g () is a linear function of log. How does this fact relate to the graph in Figure 2-8e? 122 Chapter 2: Figure 2-8e C3d. If 400 trees are planted, the population increases at first and then levels off at 00. If 1300 (too man) trees are planted, the population decreases to level off at 00. If 299 (too few) trees are planted, the population dwindles until all trees are dead. C3. Slope Field Logistic Function Problem: The logistic functions ou have studied in this chapter model populations that start at a relativel low value and then rise asmptoticall to a maimum sustainable population. There ma also be a minimum sustainable population. Suppose that a new variet of tree is planted on a relativel small island. Research indicates that the minimum sustainable population is 300 trees and that the maimum sustainable population is 00 trees. A logistic function modeling this situation is C 00 e C e 0.7 where is the number of trees alive at time, in decades after the trees were planted. The coefficients 300 and 00 are the minimum and maimum sustainable populations, respectivel, and C is a constant determined b the initial condition, the number of trees planted at time 0. a. Determine the value of C and write the particular equation if, at time 0, i. ii trees are planted. iii. 299 trees are planted. b. a window with 0 and suitable -values. What are the major differences among the three graphs? c. Figure 2-8f shows a slope field representing functions with the given equation. The line segment through each grid point indicates the slope the graph would have if it passed through that point. On a cop of Figure 2-8f, plot the three equations from part a. How are the graphs related to the line segments on the slope field? 122 Chapter 2: Properties of Elementar Functions

63 Chapter Test Figure 2-8f d. Describe the behavior of the tree population for each of the three initial conditions in part a. In particular, eplain what happens if too few trees are planted and also what happens if too man trees are planted. e. Without doing an more computations, sketch on the slope field the graph of the tree population if, at time 0, i. 00 trees had been planted. ii. 0 trees had been planted. iii. 200 trees had been planted. f. How does the slope field allow ou to analze graphicall the behavior of man related logistic functions without doing an computations? Part 1: No calculators allowed (T1 T9) c. d. T1. Write the general equation of a. A linear function b. A quadratic function c. A power function d. An eponential function e. A logarithmic function f. A logistic function T2. What tpe of function could have the graph shown? a. b. e. Section 2-8: f. T3. What numerical pattern do regularl spaced data have for a. A linear function b. A quadratic function c. An untranslated power function d. An untranslated eponential function e. An untranslated logarithmic function 123 C3e. C3f. You can draw the graph following the direction of the line segments to get an idea of what happens at different initial conditions. T1a. a 1 b T1b. a 2 1 b 1 c, a 0 T1c. a b, a 0 T1d. ae b or ab, a, b 0, b > 0 and b 1 in the case of ab T1e. a 1 b log c, b 0 and c 0, c 1 T1f. c 1 1 ae 2b or c 1 1 ab 2, a, b, c 0, b > 0 and b 1 in the case of c 1 1 ab 2 T2a. Logarithmic T2b. T2c. T2d. T2e. T2f. T3a. T3b. T3c. T3d. T3e. Eponential Logistic Quadratic Power Linear Add add Constant-second differences Multipl multipl Add multipl Multipl add Section 2-8: Chapter Review and Test 123

64 Problem Notes (continued) T4. a c b T. log log T6. 8 T7. 4 T8. 2 T9. No solutions. T. f() f() f(20) f() T11. f() a b ; f() T (1) 3 202, 0.6(20) ; the function is correct. T13. f(0) 600,000 lb 300 tons T ft T g () Graph will be concave up. The function appears to start at a positive number, decrease rapidl, and then level off as grows large. A linear function cannot work, because the graph appears to be concave. An inverse variation power function cannot work, because it appears that the graph will intersect the vertical ais. Problems T16 T18 work particularl well on a CAS. The point of these problems is not the manipulation, so students could use a sstem solver for Problem T16, use a Solve command for Problem T17, and evaluate the function directl for Problem T18. T16. f() (130.0 )( ) ; f() F f(7) F f(9) F T F above room temperature. T F above room temperature. T4. Write the equation log a b c in eponential form. T. Show how to use the logarithm of a power propert to simplif log. T6. ln 80 ln 2 ln 20 ln? T7. log 2 log 3 log? T8. Solve the equation: T9. Solve the equation: lo g 2 ( l og 2 ( 3) 8 Part 2: Graphing calculators allowed (T T28) Shark Problem: Suppose that from great white sharks caught in the past, fishermen find these weights and (ft) f () (lb) T. Show that the data set in the table has the T11. Write the general equation of a power function. Then use the points (, 7) and (, 600) to calculate algebraicall the two constants in the equation. Store these values without rounding. Write the particular equation. T12. correct b showing that it gives the other two data points in the table. T13. From fossilized shark teeth, naturalists think there were once great white sharks 0 ft long. Based on our mathematical model, how heav would such a shark be? Is this surprising? 124 Chapter 2: A quick solution for Problem T19 is shown at the right for students who know how to read the CAS output. In this case, knowing how to ask the question and interpret the answer has significant value. T14. A newspaper report describes a great white shark that weighed 3000 lb. Based on our mathematical model, about how long was the shark? Show the method ou use. Coffee Cup Problem: You pour a cup of coffee. Three F above room temperature. You record its temperature ever 2 minutes thereafter, creating this table of (ft) g () ( F above room temperature) T1. whether the graph of the function ou can fit to the points is concave up or concave down. reasonable for this function but a linear or a power function would not. T16. Find the particular equation of the eponential function that fits the points at 3 and 11. Show that the equation gives approimatel the correct values for the other three times. T17. to estimate the temperature of the coffee when it was poured. T18. Use our equation to predict the temperature of the coffee a half-hour after it was poured. T19. The Add Multipl Propert Proof Problem: 7(13 ), then log is a linear function of. T19. log log 7 1 (log 13) 124 Chapter 2: Properties of Elementar Functions