Exponential, Logistic, and Logarithmic Functions

Size: px
Start display at page:

Download "Exponential, Logistic, and Logarithmic Functions"

Transcription

1 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic Functions 3.5 Equation Solving and Modeling 3.6 Mathematics of Finance The loudness of a sound we hear is based on the intensit of the associated sound wave. This sound intensit is the energ per unit time of the wave over a given area, measured in watts per square meter 1W/m 2 2. The intensit is greatest near the source and decreases as ou move awa, whether the sound is rustling leaves or rock music. Because of the wide range of audible sound intensities, the are generall converted into decibels, which are based on logarithms. See page

2 252 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions Chapter 3 Overview In this chapter, we stud three interrelated families of functions: eponential, logistic, and logarithmic functions. Polnomial functions, rational functions, and power functions with rational eponents are algebraic functions functions obtained b adding, subtracting, multipling, and dividing constants and an independent variable, and raising epressions to integer powers and etracting roots. In this chapter and the net one, we eplore transcendental functions, which go beond, or transcend, these algebraic operations. Just like their algebraic cousins, eponential, logistic, and logarithmic functions have wide application. Eponential functions model growth and deca over time, such as unrestricted population growth and the deca of radioactive substances. Logistic functions model restricted population growth, certain chemical reactions, and the spread of rumors and diseases. Logarithmic functions are the basis of the Richter scale of earthquake intensit, the ph acidit scale, and the decibel measurement of sound. The chapter closes with a stud of the mathematics of finance, an application of eponential and logarithmic functions often used when making investments. What ou ll learn about Eponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models... and wh Eponential and logistic functions model man growth patterns, including the growth of human and animal populations FIGURE 3.1 Sketch of g12 = Eponential and Logistic Functions Eponential Functions and Their Graphs The functions ƒ12 = 2 and g12 = 2 each involve a base raised to a power, but the roles are reversed: For ƒ12 = 2, the base is the variable, and the eponent is the constant 2; ƒ is a familiar monomial and power function. For g12 = 2, the base is the constant 2, and the eponent is the variable ; g is an eponential function. See Figure 3.1. DEFINITION Eponential Functions Let a and b be real number constants. An eponential function in is a function that can be written in the form ƒ12 = a # b, where a is nonzero, b is positive, and b Z 1. The constant a is the initial value of ƒ (the value at = 0), and b is the base. Eponential functions are defined and continuous for all real numbers. It is important to recognize whether a function is an eponential function. EXAMPLE 1 Identifing Eponential Functions ƒ12 = 3 is an eponential function, with an initial value of 1 and base of 3. g12 = 6-4 is not an eponential function because the base is a variable and the eponent is a constant; g is a power function. (c) h12 = -2 # 1.5 is an eponential function, with an initial value of -2 and base of 1.5.

3 SECTION 3.1 Eponential and Logistic Functions 253 (d) k12 = 7 # 2 - is an eponential function, with an initial value of 7 and base of 1/2 because 2 - = = 11/22. (e) q12 = 5 # 6 p is not an eponential function because the eponent p is a constant; q is a constant function. Now tr Eercise 1. One wa to evaluate an eponential function, when the inputs are rational numbers, is to use the properties of eponents. EXAMPLE 2 Computing Eponential Function Values for Rational Number Inputs For ƒ12 = 2, (c) ƒ142 = 2 4 = 2 # 2 # 2 # 2 = 16 ƒ102 = 2 0 = 1 ƒ1-32 = 2-3 = = 1 8 = (d) (e) ƒa 1 2 b = 21/2 = 12 = Á ƒa b = 2-3/2 = 1 2 3/2 = 1 22 = = Á Now tr Eercise 7. There is no wa to use properties of eponents to epress an eponential function s value for irrational inputs. For eample, if ƒ12 = 2, ƒ1p2 = 2 p, but what does 2 p mean? Using properties of eponents, 2 3 = 2 # 2 # 2, = 2 31/10 = So we can find meaning for 2 p b using successivel closer rational approimations to p as shown in Table 3.1. Table 3.1 Values of ƒ12 = 2 for Rational Numbers Approaching P We can conclude that ƒ1p2 = 2 p L 8.82, which could be found directl using a grapher. The methods of calculus permit a more rigorous definition of eponential functions than we give here, a definition that allows for both rational and irrational inputs. The wa eponential functions change makes them useful in applications. This pattern of change can best be observed in tabular form.

4 254 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions ( 2, 4/9) ( 1, 4/3) ( 2, 128) [ 2.5, 2.5] b [ 10, 50] ( 1, 32) (0, 4) (0, 8) [ 2.5, 2.5] b [ 25, 150] (2, 36) (1, 12) (1, 2) (2, 1/2) FIGURE 3.2 Graphs of g12 = 4 # 3 and h12 = 8 # (1/42. (Eample 3) EXAMPLE 3 Finding an Eponential Function from Its Table of Values Determine formulas for the eponential functions g and h whose values are given in Table 3.2. Table 3.2 Values for Two Eponential Functions g12 4/9 4/ * 3 * 3 * 3 * 3 h /2 SOLUTION Because g is eponential, g12 = a # b. Because g102 = 4, the initial value a is 4. Because g112 = 4 # b 1 = 12, the base b is 3. So, Because h is eponential, h12 = a # b. Because h102 = 8, the initial value a is 8. Because h112 = 8 # b 1 = 2, the base b is 1/4. So, 8 2 g12 = 4 # 3. * 1 4 * 1 4 * 1 4 * 1 4 h12 = 8 # a 1 4 b. Figure 3.2 shows the graphs of these functions pass through the points whose coordinates are given in Table 3.2. Now tr Eercise 11. Table 3.3 Values for a General Eponential Function ƒ12 a # b a # b -2 ab -2 * b -1 ab -1 * b 0 a * b 1 ab * b 2 ab 2 Observe the patterns in the g12 and h12 columns of Table 3.2. The g12 values increase b a factor of 3 and the h12 values decrease b a factor of 1/4, as we add 1 to moving from one row of the table to the net. In each case, the change factor is the base of the eponential function. This pattern generalizes to all eponential functions as illustrated in Table 3.3. In Table 3.3, as increases b 1, the function value is multiplied b the base b. This relationship leads to the following recursive formula. Eponential Growth and Deca For an eponential function ƒ12 = a # b and an real number, ƒ1 + 1) = b # ƒ12. If a 7 0 and b 7 1, the function f is increasing and is an eponential growth function. The base b is its growth factor. If a 7 0 and b 6 1, f is decreasing and is an eponential deca function. The base b is its deca factor.

5 SECTION 3.1 Eponential and Logistic Functions 255 In Eample 3, g is an eponential growth function, and h is an eponential deca function. As increases b 1, grows b a factor of 3, and h12 = 8 # (1/42 g12 = 4 # 3 decas b a factor of 1/4. Whenever the initial value is positive, the base of an eponential function, like the slope of a linear function, tells us whether the function is increasing or decreasing and b how much. So far, we have focused most of our attention on the algebraic and numerical aspects of eponential functions. We now turn our attention to the graphs of these functions. Graphs of Eponential Functions EXPLORATION 1 1. Graph each function in the viewing window 3-2, 24 b 3-1, 64. (c) (d) 4 = 5 3 = 4 2 = 3 1 = 2 Which point is common to all four graphs? Analze the functions for domain, range, continuit, increasing or decreasing behavior, smmetr, boundedness, etrema, asmptotes, and end behavior. 2. Graph each function in the viewing window 3-2, 24 b 3-1, = a 1 1 = a 1 2 b 3 b (c) 3 = a 1 4 b (d) 4 = a 1 5 b Which point is common to all four graphs? Analze the functions for domain, range, continuit, increasing or decreasing behavior, smmetr, boundedness, etrema, asmptotes, and end behavior. We summarize what we have learned about eponential functions with an initial value of 1. f () = b b > 1 (1, b) f () = b 0 < b < 1 (0, 1) (0, 1) (1, b) FIGURE 3.3 Graphs of ƒ12 = b for b 7 1 and 0 6 b 6 1. Eponential Functions ƒ12 b Domain: All reals Range: 10, q2 Continuous No smmetr: neither even nor odd Bounded below, but not above No local etrema Horizontal asmptote: = 0 No vertical asmptotes If b 7 1 (see Figure 3.3a), then ƒ is an increasing function, lim ƒ12 = 0 and lim ƒ12 = q. : -q : q If 0 6 b 6 1 (see Figure 3.3b), then ƒ is a decreasing function, lim ƒ12 = q and lim ƒ12 = 0. : -q : q

6 256 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions The translations, reflections, stretches, and shrinks studied in Section 1.5, together with our knowledge of the graphs of basic eponential functions, allow us to predict the graphs of the functions in Eample 4. EXAMPLE 4 Transforming Eponential Functions Describe how to transform the graph of ƒ12 = 2 into the graph of the given function. Sketch the graphs b hand and support our answer with a grapher. (c) k12 = 3 # 2 g12 = 2-1 h12 = 2 - SOLUTION The graph of g12 = 2-1 is obtained b translating the graph of ƒ12 = 2 b 1 unit to the right (Figure 3.4a). We can obtain the graph of h12 = 2 - b reflecting the graph of ƒ12 = 2 across the -ais (Figure 3.4b). Because 2 - = = 11/22, we can also think of h as an eponential function with an initial value of 1 and a base of 1/2. (c) We can obtain the graph of k12 = 3 # 2 b verticall stretching the graph of ƒ12 = 2 b a factor of 3 (Figure 3.4c). Now tr Eercise 15. [ 4, 4] b [ 2, 8] [ 4, 4] b [ 2, 8] [ 4, 4] b [ 2, 8] (c) FIGURE 3.4 The graph of shown with,, and (c) k12 = 3 # 2 ƒ12 = 2 g12 = 2-1 h12 = 2 -. (Eample 4) The Natural Base e The function ƒ12 = e is one of the basic functions introduced in Section 1.3, and is an eponential growth function. BASIC FUNCTION The Eponential Function [ 4, 4] b [ 1, 5] FIGURE 3.5 The graph of ƒ12 = e. ƒ12 = e Domain: All reals Range: 10, q2 Continuous Increasing for all No smmetr Bounded below, but not above No local etrema Horizontal asmptote: = 0 No vertical asmptotes End behavior: lim and lim : q e = q : -q e = 0 Because ƒ12 = e is increasing, it is an eponential growth function, so e 7 1. But what is e, and what makes this eponential function the eponential function?

7 SECTION 3.1 Eponential and Logistic Functions 257 The letter e is the initial of the last name of Leonhard Euler ( ), who introduced the notation. Because ƒ12 = e has special calculus properties that simplif man calculations, e is the natural base of eponential functions for calculus purposes, and ƒ12 = e is considered the natural eponential function. DEFINITION The Natural Base e e = lim a1 + 1 : q b We cannot compute the irrational number e directl, but using this definition we can obtain successivel closer approimations to e, as shown in Table 3.4. Continuing the process in Table 3.4 with a sufficientl accurate computer can show that e L f () = e k k > 0 (0, 1) (1, e k ) Table 3.4 Approimations Approaching the Natural Base e , , / We are usuall more interested in the eponential function ƒ12 = e and variations of this function than in the irrational number e. In fact, an eponential function can be epressed in terms of the natural base e. f () = e k k < 0 THEOREM Eponential Functions and the Base e An eponential function ƒ12 = a # b can be rewritten as ƒ12 = a # e k, (0, 1) (1, e k ) for an appropriatel chosen real number constant k. If and k 7 0, ƒ12 = a # e k a 7 0 is an eponential growth function. (See Figure 3.6a.) If and k 6 0, ƒ12 = a # e k a 7 0 is an eponential deca function. (See Figure 3.6b.) FIGURE 3.6 Graphs of ƒ12 = e k for k 7 0 and k 6 0. In Section 3.3 we will develop some mathematics so that, for an positive number b Z 1, we can easil find the value of k such that e k = b. In the meantime, we can use graphical and numerical methods to approimate k, as ou will discover in Eploration 2. EXPLORATION 2 Choosing k so that e k 2 1. Graph ƒ12 = 2 in the viewing window 3-4, 44 b 3-2, One at a time, overla the graphs of g12 = e k for k = 0.4, 0.5, 0.6, 0.7, and 0.8. For which of these values of k does the graph of g most closel match the graph of ƒ? 3. Using tables, find the 3-decimal-place value of k for which the values of g most closel approimate the values of ƒ.

8 258 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions EXAMPLE 5 Transforming Eponential Functions Describe how to transform the graph of ƒ12 = e into the graph of the given function. Sketch the graphs b hand and support our answer with a grapher. g12 = e 2 h12 = e - (c) k12 = 3e [ 4, 4] b [ 2, 8] SOLUTION The graph of g12 = e 2 is obtained b horizontall shrinking the graph of ƒ12 = e b a factor of 2 (Figure 3.7a). We can obtain the graph of h12 = e - b reflecting the graph of ƒ12 = e across the -ais (Figure 3.7b). (c) We can obtain the graph of k12 = 3e b verticall stretching the graph of ƒ12 = e b a factor of 3 (Figure 3.7c). Now tr Eercise 21. [ 4, 4] b [ 2, 8] [ 4, 4] b [ 2, 8] (c) FIGURE 3.7 The graph of ƒ12 = e shown with g12 = e 2, h12 = e -, and (c) k12 = 3e. (Eample 5) Logistic Functions and Their Graphs Eponential growth is unrestricted. An eponential growth function increases at an ever-increasing rate and is not bounded above. In man growth situations, however, there is a limit to the possible growth. A plant can onl grow so tall. The number of goldfish in an aquarium is limited b the size of the aquarium. In such situations the growth often begins in an eponential manner, but the growth eventuall slows and the graph levels out. The associated growth function is bounded both below and above b horizontal asmptotes. DEFINITION Logistic Growth Functions Let a, b, c, and k be positive constants, with b 6 1. A logistic growth function in is a function that can be written in the form ƒ12 = c 1 + a # b or ƒ12 = where the constant c is the limit to growth. c 1 + a # e -k Aliases for Logistic Growth Logistic growth is also known as restricted, inhibited, or constrained eponential growth. If b 7 1 or k 6 0, these formulas ield logistic deca functions. Unless otherwise stated, all logistic functions in this book will be logistic growth functions. B setting a = c = k = 1, we obtain the logistic function ƒ12 = e -. This function, though related to the eponential function e, cannot be obtained from b translations, reflections, and horizontal and vertical stretches and shrinks. So we give the logistic function a formal introduction: e

9 SECTION 3.1 Eponential and Logistic Functions 259 BASIC FUNCTION The Logistic Function [ 4.7, 4.7] b [ 0.5, 1.5] FIGURE 3.8 The graph of ƒ12 = 1/11 + e ƒ12 = 1 + e - Domain: All reals Range: 10, 12 Continuous Increasing for all Smmetric about 10, 1/22, but neither even nor odd Bounded below and above No local etrema Horizontal asmptotes: = 0 and = 1 No vertical asmptotes End behavior: lim ƒ12 = 0 and lim ƒ12 = 1 : -q : q All logistic growth functions have graphs much like the basic logistic function. Their end behavior is alwas described b the equations lim ƒ12 = 0 : -q and lim ƒ12 = c, : q where c is the limit to growth (see Eercise 73). All logistic functions are bounded b their horizontal asmptotes, = 0 and = c, and have a range of 10, c2. Although ever logistic function is smmetric about the point of its graph with - coordinate c/2, this point of smmetr is usuall not the -intercept, as we can see in Eample 6. EXAMPLE 6 Graphing Logistic Growth Functions Graph the function. Find the -intercept and the horizontal asmptotes. ƒ12 = # 0.7 SOLUTION The graph of ƒ12 = 8/ # is shown in Figure 3.9a. The -intercept is Because the limit to growth is 8, the horizontal asmptotes are = 0 and = 8. The graph of g12 = 20/11 + 2e -3 2 is shown in Figure 3.9b. The -intercept is g102 = ƒ102 = g12 = # = = e -3 # 0 = e = 20/3 L Because the limit to growth is 20, the horizontal asmptotes are = 0 and = 20. Now tr Eercise 41.

10 260 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions [ 10, 20] b [ 2, 10] [ 2, 4] b [ 5, 25] FIGURE 3.9 The graphs of ƒ12 = 8/(1 + 3 # and g12 = 20/(1 + 2e (Eample 6) Population Models Eponential and logistic functions have man applications. One area where both tpes of functions are used is in modeling population. Between 1990 and 2000, both Phoeni and San Antonio passed the 1 million mark. With its Silicon Valle industries, San Jose, California, appears to be the net U.S. cit destined to surpass 1 million residents. When a cit s population is growing rapidl, as in the case of San Jose, eponential growth is a reasonable model. A Note on Population Data When the U.S. Census Bureau reports a population estimate for a given ear, it generall represents the population at the middle of the ear, or Jul 1. We will assume this to be the case when interpreting our results to population problems unless otherwise noted. EXAMPLE 7 Modeling San Jose s Population Using the data in Table 3.5 and assuming the growth is eponential, when will the population of San Jose, California, surpass 1 million persons? SOLUTION Model Let P1t2 be the population of San Jose t ears after Jul 1, (See margin note.) Because P is eponential, P1t2 = P 0 # b t, where P 0 is the initial (2000) population of 898,759. From Table 3.5 we see that P172 = b 7 = So, b = B 7 939, ,759 L and P1t2 = 898,759 # t. Solve Graphicall Figure 3.10 shows that this population model intersects = 1,000,000 when the independent variable is about Interpret Because r after mid-2000 is in the first half of 2017, according to this model the population of San Jose will surpass the 1 million mark in earl Now tr Eercise 51. Table 3.5 The Population of San Jose, California Year Population , ,899 Source: U.S. Census Bureau. Intersection X= Y= [ 10, 30] b [ , ] FIGURE 3.10 A population model for San Jose, California. (Eample 7)

11 SECTION 3.1 Eponential and Logistic Functions 261 While San Jose s population is soaring, in other major cities, such as Dallas, the population growth is slowing. The once sprawling Dallas is now constrained b its neighboring cities. A logistic function is often an appropriate model for restricted growth, such as the growth that Dallas is eperiencing. Intersection X= Y= EXAMPLE 8 Modeling Dallas s Population Based on recent census data, a logistic model for the population of Dallas, t ears after 1900, is as follows: P1t2 = 1,301, e t According to this model, when was the population 1 million? SOLUTION Figure 3.11 shows that the population model intersects = 1,000,000 when the independent variable is about Because r after mid-1900 is at the beginning of 1985, if Dallas s population has followed this logistic model, its population was 1 million then. Now tr Eercise 55. [0, 120] b [ , ] FIGURE 3.11 A population model for Dallas, Teas. (Eample 8) QUICK REVIEW 3.1 (For help, go to Sections A.1 and P.1.) Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 1 4, evaluate the epression without using a calculator A / /2 In Eercises 5 8, rewrite the epression using a single positive eponent a b In Eercises 9 10, use a calculator to evaluate the epression SECTION 3.1 EXERCISES In Eercises 1 6, which of the following are eponential functions? For those that are eponential functions, state the initial value and the base. For those that are not, eplain wh not. 1. = 8 2. = 3 3. = 5 4. = = 2 6. = 1.3 In Eercises 7 10, compute the eact value of the function for the given -value without using a calculator. 7. ƒ12 = 3 # 5 for = 0 8. ƒ12 = 6 # 3 for = ƒ12 = -2 # 3 for = 1/3 10. ƒ12 = 8 # 4 for = -3/2 In Eercises 11 and 12, determine a formula for the eponential function whose values are given in Table ƒ g12 Table 3.6 Values for Two Eponential Functions ƒ12 g / / /8 4/3

12 262 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions In Eercises 13 and 14, determine a formula for the eponential function whose graph is shown in the figure. 13. ƒ g12 = f() (0, 3) (2, 6) = g() (0, 2) b a1, 2 e In Eercises 15 24, describe how to transform the graph of ƒ into the graph of g. Sketch the graphs b hand and support our answer with a grapher ƒ12 = 2, g12 = ƒ12 = 3, g12 = ƒ12 = 4, g12 = ƒ12 = 2, g12 = ƒ12 = 0.5, g12 = 3 # , g12 = 2 # ƒ12 = ƒ12 = e, g12 = e ƒ12 = e, g12 = -e ƒ12 = e, g12 = 2e 24. ƒ12 = e, g12 = 3e 2-1 In Eercises 25 30, match the given function with its graph. Writing to Learn Eplain how to make the choice without using a grapher. 25. = = = = = = In Eercises 31 34, state whether the function is an eponential growth function or eponential deca function, and describe its end behavior using limits. 31. ƒ12 = ƒ12 = a 1 e b 33. ƒ12 = ƒ12 = In Eercises 35 38, solve the inequalit graphicall a 1 4 b 7 a 1 3 b 38. a 1 3 b 6 a 1 2 b Group Activit In Eercises 39 and 40, use the properties of eponents to prove that two of the given three eponential functions are identical. Support graphicall = = (c) 3 = = = (c) 3 = In Eercises 41 44, use a grapher to graph the function. Find the -intercept and the horizontal asmptotes ƒ12 = 42. ƒ12 = # # ƒ12 = 44. g12 = 1 + 3e e - In Eercises 45 50, graph the function and analze it for domain, range, continuit, increasing or decreasing behavior, smmetr, boundedness, etrema, asmptotes, and end behavior ƒ12 = 4 # 0.5 ƒ12 = 3 # ƒ12 = 5 # e - ƒ12 = 4 # e ƒ12 = 50. ƒ12 = # e # e Population Growth Using the midear data in Table 3.7 and assuming the growth is eponential, when did the population of Austin surpass 800,000 persons? (c) (e) (d) (f) Table 3.7 Populations of Two Major U.S. Cities Cit 1990 Population 2000 Population Austin, Teas 465, ,562 Columbus, Ohio 632, ,265 Source: World Almanac and Book of Facts 2005.

13 SECTION 3.1 Eponential and Logistic Functions Population Growth Using the data in Table 3.7 and assuming the growth is eponential, when would the population of Columbus surpass 800,000 persons? 53. Population Growth Using the data in Table 3.7 and assuming the growth is eponential, when were the populations of Austin and Columbus equal? 54. Population Growth Using the data in Table 3.7 and assuming the growth is eponential, which cit Austin or Columbus would reach a population of 1 million first, and in what ear? 55. Population Growth Using 20th-centur U.S. census data, the population of Ohio can be modeled b P1t2 = e , where P is the population in millions and t is the number of ears since April 1, Based on this model, when was the population of Ohio 10 million? 56. Population Growth Using 20th-centur U.S. census data, the population of New York state can be modeled b P1t2 = e t, where P is the population in millions and t is the number of ears since Based on this model, What was the population of New York in 1850? What will New York state s population be in 2015? (c) What is New York s maimum sustainable population (limit to growth)? 57. Bacteria Growth The number B of bacteria in a petri dish culture after t hours is given b B = 100e 0.693t. What was the initial number of bacteria present? How man bacteria are present after 6 hours? 58. Carbon Dating The amount C in grams of carbon-14 present in a certain substance after t ears is given b C = 20e t. What was the initial amount of carbon-14 present? How much is left after 10,400 ears? When will the amount left be 10 g? Standardized Test Questions 59. True or False Ever eponential function is strictl increasing. Justif our answer. 60. True or False Ever logistic growth function has two horizontal asmptotes. Justif our answer. In Eercises 61 64, solve the problem without using a calculator. 61. Multiple Choice Which of the following functions is eponential? 3 (A) ƒ12 = a 2 (B) ƒ12 = (C) ƒ12 = 2/3 (D) ƒ12 = 23 (E) ƒ12 = Multiple Choice What point do all functions of the form ƒ12 = b 1b 7 02 have in common? (A) 11, 12 (B) 11, 02 (C) 10, 12 (D) 10, 02 (E) 1-1, Multiple Choice The growth factor for ƒ12 = 4 # 3 is (A) 3. (B) 4. (C) 12. (D) 64. (E) Multiple Choice For 7 0, which of the following is true? (A) (B) (C) 11/62 7 (1/22 (D) (E) Eplorations 65. Graph each function and analze it for domain, range, increasing or decreasing behavior, boundedness, etrema, asmptotes, and end behavior. ƒ12 = # e g12 = e- 66. Use the properties of eponents to solve each equation. Support graphicall. 2 = = 27 (c) 8 /2 = 4 +1 (d) 9 = 3 +1 Etending the Ideas 67. Writing to Learn Table 3.8 gives function values for = ƒ12 and = g12. Also, three different graphs are shown. Table 3.8 Data for Two Functions ƒ12 g Which curve of those shown in the graph most closel resembles the graph of = ƒ12? Eplain our choice. Which curve most closel resembles the graph of = g12? Eplain our choice.

14 264 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 68. Writing to Learn Let ƒ12 = 2. Eplain wh the graph of ƒ1a + b2 can be obtained b appling one transformation to the graph of = c for an appropriate value of c. What is c? Eercises refer to the epression ƒ1a, b, c2 = a # b c. For eample, if,, and, the epression is ƒ12, 3, 2 = 2 # 3 a = 2 b = 3 c =, an eponential function. 69. If b =, state conditions on a and c under which the epression ƒ1a, b, c2 is a quadratic power function. 70. If b =, state conditions on a and c under which the epression ƒ(a, b, c) is a decreasing linear function. 71. If c =, state conditions on a and b under which the epression ƒ(a, b, c) is an increasing eponential function. 72. If c =, state conditions on a and b under which the epression ƒ(a, b, c) is a decreasing eponential function. 73. Prove that lim c c and lim, : q 1 + a # b = c : -q 1 + a # b = 0 for constants a, b, and c, with a 7 0, 0 6 b 6 1, and c 7 0.

3.3 Logarithmic Functions and Their Graphs

3.3 Logarithmic Functions and Their Graphs 274 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions What ou ll learn about Inverses of Eponential Functions Common Logarithms Base 0 Natural Logarithms Base e Graphs of Logarithmic Functions

More information

PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1: Exponential and Logistic Functions

PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1: Exponential and Logistic Functions PRE-CALCULUS: Finne,Demana,Watts and Kenned Chapter 3: Eponential, Logistic, and Logarithmic Functions 3.1: Eponential and Logistic Functions Which of the following are eponential functions? For those

More information

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic

More information

3.2. Exponential and Logistic Modeling. Finding Growth and Decay Rates. What you ll learn about

3.2. Exponential and Logistic Modeling. Finding Growth and Decay Rates. What you ll learn about 290 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.2 Eponential and Logistic Modeling What you ll learn about Constant Percentage Rate and Eponential Functions Eponential Growth and Decay

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions 7 Eponential and Logarithmic Functions In this chapter ou will stud two tpes of nonalgebraic functions eponential functions and logarithmic functions. Eponential and logarithmic functions are widel used

More information

1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs

1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs 0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals

More information

Exponential functions are defined and for all real numbers.

Exponential functions are defined and for all real numbers. 3.1 Exponential and Logistic Functions Objective SWBAT evaluate exponential expression and identify and graph exponential and logistic functions. Exponential Function Let a and b be real number constants..

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

More information

6.4 graphs OF logarithmic FUnCTIOnS

6.4 graphs OF logarithmic FUnCTIOnS SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS

More information

The Natural Base e. ( 1, e 1 ) 220 Chapter 3 Exponential and Logarithmic Functions. Example 6 Evaluating the Natural Exponential Function.

The Natural Base e. ( 1, e 1 ) 220 Chapter 3 Exponential and Logarithmic Functions. Example 6 Evaluating the Natural Exponential Function. 0 Chapter Eponential and Logarithmic Functions (, e) f() = e (, e ) (0, ) (, e ) FIGURE.9 The Natural Base e In man applications, the most convenient choice for a base is the irrational number e.78888....

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs .1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic

More information

Practice A ( 1, 3 ( 0, 1. Match the function with its graph. 3 x. Explain how the graph of g can be obtained from the graph of f. 5 x.

Practice A ( 1, 3 ( 0, 1. Match the function with its graph. 3 x. Explain how the graph of g can be obtained from the graph of f. 5 x. 8. Practice A For use with pages 65 7 Match the function with its graph.. f. f.. f 5. f 6. f f Lesson 8. A. B. C. (, 6) (0, ) (, ) (0, ) ( 0, ) (, ) D. E. F. (0, ) (, 6) ( 0, ) (, ) (, ) (0, ) Eplain how

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Eponential and Logarithmic Equations.5 Eponential and Logarithmic

More information

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. 1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function

More information

f 0 ab a b: base f

f 0 ab a b: base f Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential

More information

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of

More information

8.1 Exponents and Roots

8.1 Exponents and Roots Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents

More information

5A Exponential functions

5A Exponential functions Chapter 5 5 Eponential and logarithmic functions bjectives To graph eponential and logarithmic functions and transformations of these functions. To introduce Euler s number e. To revise the inde and logarithm

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs

More information

1.2 Functions and Their Properties PreCalculus

1.2 Functions and Their Properties PreCalculus 1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Eponential and Logarithmic Equations.5 Eponential and Logarithmic

More information

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator,

a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI-83 Graphing Calculator, GRADE PRE-CALCULUS UNIT C: QUADRATIC FUNCTIONS CLASS NOTES FRAME. After linear functions, = m + b, and their graph the Quadratic Functions are the net most important equation or function. The Quadratic

More information

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +

More information

a > 0 parabola opens a < 0 parabola opens

a > 0 parabola opens a < 0 parabola opens Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(

More information

1. For each of the following, state the domain and range and whether the given relation defines a function. b)

1. For each of the following, state the domain and range and whether the given relation defines a function. b) Eam Review Unit 0:. For each of the following, state the domain and range and whether the given relation defines a function. (,),(,),(,),(5,) a) { }. For each of the following, sketch the relation and

More information

CHAPTER 2 Polynomial and Rational Functions

CHAPTER 2 Polynomial and Rational Functions CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............

More information

Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)

Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.) Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,

More information

8-1 Exploring Exponential Models

8-1 Exploring Exponential Models 8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y =

More information

Sections 4.1 & 4.2 Exponential Growth and Exponential Decay

Sections 4.1 & 4.2 Exponential Growth and Exponential Decay 8 Sections 4. & 4.2 Eponential Growth and Eponential Deca What You Will Learn:. How to graph eponential growth functions. 2. How to graph eponential deca functions. Eponential Growth This is demonstrated

More information

Logarithms. Bacteria like Staph aureus are very common.

Logarithms. Bacteria like Staph aureus are very common. UNIT 10 Eponentials and Logarithms Bacteria like Staph aureus are ver common. Copright 009, K1 Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,

More information

P.4 Lines in the Plane

P.4 Lines in the Plane 28 CHAPTER P Prerequisites P.4 Lines in the Plane What ou ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

decreases as x increases.

decreases as x increases. Chapter Review FREQUENTLY ASKED Questions Q: How can ou identif an eponential function from its equation? its graph? a table of values? A: The eponential function has the form f () 5 b, where the variable

More information

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

More information

LESSON 12.2 LOGS AND THEIR PROPERTIES

LESSON 12.2 LOGS AND THEIR PROPERTIES LESSON. LOGS AND THEIR PROPERTIES LESSON. LOGS AND THEIR PROPERTIES 5 OVERVIEW Here's what ou'll learn in this lesson: The Logarithm Function a. Converting from eponents to logarithms and from logarithms

More information

f 0 ab a b: base f

f 0 ab a b: base f Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential

More information

c) domain {x R, x 3}, range {y R}

c) domain {x R, x 3}, range {y R} Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..

More information

) approaches e

) approaches e COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural

More information

7Exponential and. Logarithmic Functions

7Exponential and. Logarithmic Functions 7Eponential and Logarithmic Functions A band of green light occasionall appears above the rising or setting sun. This phenomenon is known as a green flash because it lasts for a ver brief period of time.

More information

Explore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x.

Explore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x. 1_ 8 6 8 Locker LESSON 13. The Base e Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes of the graphs of ƒ () = b and ƒ () = log b () where b is, 1, and e

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

7.4. Characteristics of Logarithmic Functions with Base 10 and Base e. INVESTIGATE the Math

7.4. Characteristics of Logarithmic Functions with Base 10 and Base e. INVESTIGATE the Math 7. Characteristics of Logarithmic Functions with Base 1 and Base e YOU WILL NEED graphing technolog EXPLORE Use benchmarks to estimate the solution to this equation: 1 5 1 logarithmic function A function

More information

Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions. Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and

More information

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1 Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

More information

15.2 Graphing Logarithmic

15.2 Graphing Logarithmic _ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b

More information

3.4. Properties of Logarithmic Functions

3.4. Properties of Logarithmic Functions 310 CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.4 Properties of Logarithmic Functions What you ll learn about Properties of Logarithms Change of Base Graphs of Logarithmic Functions with

More information

5.4 Logarithmic Functions

5.4 Logarithmic Functions SECTION 5.4 Logarithmic Functions 8 Problems and provide definitions for two other transcendental functions.. The hperbolic sine function, designated b sinh, is defined as sinh = e - e - (a) Show that

More information

1.3 Exponential Functions

1.3 Exponential Functions Section. Eponential Functions. Eponential Functions You will be to model eponential growth and decay with functions of the form y = k a and recognize eponential growth and decay in algebraic, numerical,

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions.1 Eponential Growth and Deca Functions. The Natural Base e.3 Logarithms and Logarithmic Functions. Transformations of Eponential and Logarithmic Functions.5 Properties

More information

QUADRATIC FUNCTION REVIEW

QUADRATIC FUNCTION REVIEW Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important

More information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

Algebra 2 Honors Summer Packet 2018

Algebra 2 Honors Summer Packet 2018 Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

More information

Summary, Review, and Test

Summary, Review, and Test 45 Chapter Equations and Inequalities Chapter Summar Summar, Review, and Test DEFINITIONS AND CONCEPTS EXAMPLES. Eponential Functions a. The eponential function with base b is defined b f = b, where b

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Linear Equations and Arithmetic Sequences

Linear Equations and Arithmetic Sequences CONDENSED LESSON.1 Linear Equations and Arithmetic Sequences In this lesson, ou Write eplicit formulas for arithmetic sequences Write linear equations in intercept form You learned about recursive formulas

More information

Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!

Name Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit! Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).

More information

Ready To Go On? Skills Intervention 6-1 Polynomials

Ready To Go On? Skills Intervention 6-1 Polynomials 6A Read To Go On? Skills Intervention 6- Polnomials Find these vocabular words in Lesson 6- and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading

More information

Chapter 2 Polynomial, Power, and Rational Functions

Chapter 2 Polynomial, Power, and Rational Functions Section. Linear and Quadratic Functions and Modeling 6 Chapter Polnomial, Power, and Rational Functions Section. Linear and Quadratic Functions and Modeling Eploration. $000 per ear.. The equation will

More information

1.7 Inverse Functions

1.7 Inverse Functions 71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,

More information

Chapter 9 Vocabulary Check

Chapter 9 Vocabulary Check 9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section 9.. 67. f = + 68. f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population

More information

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a: SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert

More information

Cubic and quartic functions

Cubic and quartic functions 3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving

More information

Ready To Go On? Skills Intervention 7-1 Exponential Functions, Growth, and Decay

Ready To Go On? Skills Intervention 7-1 Exponential Functions, Growth, and Decay 7A Find these vocabular words in Lesson 7-1 and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention 7-1 Eponential Functions, Growth, and Deca eponential growth eponential deca asmptote

More information

Name Date. Work with a partner. Each graph shown is a transformation of the parent function

Name Date. Work with a partner. Each graph shown is a transformation of the parent function 3. Transformations of Eponential and Logarithmic Functions For use with Eploration 3. Essential Question How can ou transform the graphs of eponential and logarithmic functions? 1 EXPLORATION: Identifing

More information

Chapter 3. Exponential and Logarithmic Functions. Selected Applications

Chapter 3. Exponential and Logarithmic Functions. Selected Applications Chapter 3 Eponential and Logarithmic Functions 3. Eponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Eponential and Logarithmic Equations

More information

PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.

PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line. MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for

More information

7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers.

7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers. 7-1 Eploring Eponential Models Vocabular Review 1. Cross out the epressions that are NOT powers. 16 6a 1 7. Circle the eponents in the epressions below. 5 6 5a z Vocabular Builder eponential deca (noun)

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.

More information

Exponential and Logarithmic Functions, Applications, and Models

Exponential and Logarithmic Functions, Applications, and Models 86 Eponential and Logarithmic Functions, Applications, and Models Eponential Functions In this section we introduce two new tpes of functions The first of these is the eponential function Eponential Function

More information

2 nd Semester Final Exam Review Block Date

2 nd Semester Final Exam Review Block Date Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. 1 (10-1) 1. (10-1). (10-1) 3. (10-1) 4. 3 Graph each function. Identif the verte, ais of smmetr, and

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define

More information

We all learn new things in different ways. In. Properties of Logarithms. Group Exercise. Critical Thinking Exercises

We all learn new things in different ways. In. Properties of Logarithms. Group Exercise. Critical Thinking Exercises Section 4.3 Properties of Logarithms 437 34. Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions 7 Eponential and Logarithmic Functions 7.1 Eponential Growth and Deca Functions 7. The Natural Base e 7.3 Logarithms and Logarithmic Functions 7. Transformations of Eponential and Logarithmic Functions

More information

2 nd Semester Final Exam Review Block Date

2 nd Semester Final Exam Review Block Date Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict

More information

Unit 10 - Graphing Quadratic Functions

Unit 10 - Graphing Quadratic Functions Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

More information

2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits

2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits . Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of

More information

College Algebra Final, 7/2/10

College Algebra Final, 7/2/10 NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational

More information

Algebra 2 CPA Summer Assignment 2018

Algebra 2 CPA Summer Assignment 2018 Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

More information

4.3 Mean-Value Theorem and Monotonicity

4.3 Mean-Value Theorem and Monotonicity .3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such

More information

Vocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient.

Vocabulary. Term Page Definition Clarifying Example degree of a monomial. degree of a polynomial. end behavior. leading coefficient. CHAPTER 6 Vocabular The table contains important vocabular terms from Chapter 6. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. Term Page Definition Clarifing

More information

5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS

5.6 RATIOnAl FUnCTIOnS. Using Arrow notation. learning ObjeCTIveS CHAPTER PolNomiAl ANd rational functions learning ObjeCTIveS In this section, ou will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identif

More information

Attributes of Polynomial Functions VOCABULARY

Attributes of Polynomial Functions VOCABULARY 8- Attributes of Polnomial Functions TEKS FCUS Etends TEKS ()(A) Graph the functions f () =, f () =, f () =, f () =, f () = b, f () =, and f () = log b () where b is,, and e, and, when applicable, analze

More information

2.1 Evaluate and Graph Polynomial

2.1 Evaluate and Graph Polynomial 2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of

More information

Functions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f.

Functions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f. 7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A Eponential Growth and Deca Functions Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing

More information

Lesson 5.1 Exponential Functions

Lesson 5.1 Exponential Functions Lesson.1 Eponential Functions 1. Evaluate each function at the given value. Round to four decimal places if necessar. a. r (t) 2(1 0.0) t, t 8 b. j() 9.(1 0.09), 10 2. Record the net three terms for each

More information

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals

Math 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (

More information

( 1 3, 3 4 ) Selected Answers SECTION P.2. Quick Review P.1. Exercises P.1. Quick Review P.2. Exercises P.2 SELECTED ANSWERS 895

( 1 3, 3 4 ) Selected Answers SECTION P.2. Quick Review P.1. Exercises P.1. Quick Review P.2. Exercises P.2 SELECTED ANSWERS 895 44_Demana_SEANS_pp89-04 /4/06 8:00 AM Page 89 SELECTED ANSWERS 89 Selected Answers Quick Review P.. {,,, 4,, 6}. {,, }. (a) 87.7 (b) 4.7 7. ;.7 9. 0,,,, 4,, 6 Eercises P.. 4.6 (terminating)..6 (repeating).

More information

where a 0 and the base b is a positive number other

where a 0 and the base b is a positive number other 7. Graph Eponential growth functions No graphing calculators!!!! EXPONENTIAL FUNCTION A function of the form than one. a b where a 0 and the base b is a positive number other a = b = HA = Horizontal Asmptote:

More information

Math 121. Practice Problems from Chapter 4 Fall 2016

Math 121. Practice Problems from Chapter 4 Fall 2016 Math 11. Practice Problems from Chapter Fall 01 1 Inverse Functions 1. The graph of a function f is given below. On same graph sketch the inverse function of f; notice that f goes through the points (0,

More information

Functions. Introduction

Functions. Introduction Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of

More information

CHAPTER 3 Exponential and Logarithmic Functions

CHAPTER 3 Exponential and Logarithmic Functions CHAPTER Eponential and Logarithmic Functions Section. Eponential Functions and Their Graphs......... Section. Logarithmic Functions and Their Graphs......... Section. Properties of Logarithms..................

More information

Section 2.5: Graphs of Functions

Section 2.5: Graphs of Functions Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree

More information

( 3x. Chapter Review. Review Key Vocabulary. Review Examples and Exercises 6.1 Properties of Square Roots (pp )

( 3x. Chapter Review. Review Key Vocabulary. Review Examples and Exercises 6.1 Properties of Square Roots (pp ) 6 Chapter Review Review Ke Vocabular closed, p. 266 nth root, p. 278 eponential function, p. 286 eponential growth, p. 296 eponential growth function, p. 296 compound interest, p. 297 Vocabular Help eponential

More information

3.2 Logarithmic Functions and Their Graphs

3.2 Logarithmic Functions and Their Graphs 96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that

More information

Unit 3 Notes Mathematical Methods

Unit 3 Notes Mathematical Methods Unit 3 Notes Mathematical Methods Foundational Knowledge Created b Triumph Tutoring Copright info Copright Triumph Tutoring 07 Triumph Tutoring Pt Ltd ABN 60 607 0 507 First published in 07 All rights

More information

Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1

Learning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1 College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;

More information

Properties of Elementary Functions

Properties of Elementary Functions 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

More information