1 Section 10 Exponential Functions We now turn our attention back to exponential functions In order to work with these functions effectively, it is important that we know how exponents can be manipulated This, in turn, requires an understanding of roots So, this is where we begin Part 1 Roots Example 1: Complete the following table without using a calculator (You don t need to complete the gray cells) 4 8 16 32 64 4-8 16-32 64 9 27 81 9-27 81 16 64 16-64 Example 2: Determine the missing numbers in each of the following equations The missing number is 3 There are two possible missing numbers, 3 and -3 There are two possible missing numbers, 8 and -8 The missing number is 2 (e) There is no real number which, when squared, equals -16 (A real number is a number that you would find on a number line When you square a real number, the result is either positive or zero) The missing numbers you were looking for in the last example are called roots For example, 3 is a cube root of 27, 8 and -8 are square roots of 64, and -2 is a fifth root of -32 In general, an nth root of a number a is the number which, when raised to the nth power, equals a In other words,
2 Example 3: Find indicated roots of the following numbers square roots of 36 Since, 6 and -6 are both square roots of 36 cube roots of 8 Since, 2 is a cube root of 8 4th roots of 81 Since, 3 and -3 are both 4th roots of 81 cube roots of -64 Since, -4 is a cube root of -64 Notation for Roots: We denote the nth root of a number a by The symbol is called a radical symbol For even-numbered roots (ie square roots, fourth roots, etc), refers only to the positive root For square roots, we write rather than Example 4: Find the indicated roots: since Both 5 and -5 are square roots of 25 However, refers to the positive square root So, Both 2 and -2 are fourth roots of 16 However, refers to the positive fourth root So, does not exist (as a real number) since it is not possible to square a real number and get -81 (e) since
3 Part 2 Properties of Exponents We begin with the five most basic properties of exponents: Motivating Example Property Example 5: Simplify the following expressions using the properties above (e)
4 The next three properties might appear less intuitive than the properties on the previous page, but they are the logical consequences of these properties Motivating Example Property On the one hand, On the other hand, So, On the one hand, On the other hand, So, So, Example 6: Simplify each of the following expressions using the above properties (e) Example 7: Use the exponent properties that you ve learned today to simply the following expressions
5 (e) (f) Example 8: Write using exponents is not equal to an integer (ie like -3, -2, -1, 0, 1, 2, 3, ) Determine a decimal value (approximation) using your graphing calculator Type 9^(1/4) into your calculator (The parentheses are necessary! Why do you think this is?) The numerical value returned is 173205 Part 3 Return to Exponential Growth and Decay Example 9: Suppose a population grows 6% every 3 years Determine the annual percentage growth The most obvious thing to try is to take Let s check to see if this is correct If the population is actually growing by 2% annually, then it must be growing by a factor of 102 each year Use this factor to complete the table Don t round your values! Year 0 1 2 3 Population 500 510 5202 530604 Now, did the population increase by 6% over these 3 years? (ie Did it grow by a factor of 106?) 530604/500 = 1061208 So, the population actually increased by 61208%!! Clearly, dividing 6% by 3 years was not the correct way to determine the annual percentage growth So, what is the correct way? Let R represent the annual growth factor It follows that Why is this? Well, to obtain a 3-year factor, you would need to multiply by the 1-year factor three times So, then, what must
6 R equal? From our earlier work, If we evaluate this using a calculator, we find that R = 10196128 So, according to this calculation, the population would be growing by approximately 196% each year, which is a little less than 2% Let s recalculate the table on the previous page to make sure that this factor is correct Year 0 1 2 3 Population 500 509806 519805 530 Now, if we divide 530 by 500, we see that the population grew by a factor of 106 (6%) over the three years The results from this example are summarized below If a quantity is growing or decaying exponentially by a factor of c over n-unit time periods, then the quantity is growing by a factor of over 1-unit time periods Example 10: Assume the quantities below grow or decay exponentially Determine the growth/decay factors and the corresponding percentage growth/decay over 1-unit time periods Quantity grows by 45% every 10 years The 10-year growth factor is 145 So, the annual growth factor = In other words, the quantity grows 38% annually Quantity decays by 21% every 3 weeks The 3-week decay factor is 079 So, the weekly decay factor = corresponds to a decay of 8% each week Quantity grows by 26% every 10 days The 10-day growth factor is 126 So, the daily growth factor = corresponds to a growth of 23% each day which which Example 11: Shown below are some exponential functions Use exponent properties to write the functions in the form Note: This shows that growing by a factor of 8 during 3-unit periods of time is equivalent to growing by a factor of 2 over 1-unit periods of time
7 This shows that growing by a factor of 81 during 4-unit periods of time is equivalent to growing by a factor of 3 over 1-unit periods of time This shows that growing by a factor of 157 during 7-unit periods of time is equivalent to growing by a factor of 106656 over 1-unit periods of time Example 12: In 1999, global consumption of bottled water was approximately 26 billion gallons By 2004, consumption reached 41 billion gallons (Bottled Water: Pouring Resources Down the Drain Earth Policy Institute (2006), http://wwwearthpolicyorg) By what percentage did consumption of bottled water increase during this period? The 5-year growth factor is 41/26 = 1577 Thus, consumption grew 577% during this period What was the average annual percentage growth during this period? Since the 5-year growth factor is 1577, the average annual growth factor is, which corresponds to 954% average annual growth during this period (Note: We never assumed that consumption of bottled water was growing exponentially The real annual percentage growth probably varies from year to year The 954% is therefore only an average measure of growth during this period) If we assume that bottled water consumption is growing exponentially, determine two different function equations describing this growth Let t represent time in years since 1999 and let y represent consumption (in billions of gallons) We begin with the basic equation for an exponential function Recall that b represents the initial amount Since t = 0 in the year 1999, b = 26 billion gallons In determining values for c and m, we now have a choice: Either choose c = 1577 and m = 5, or choose c = 10954 and n = 1 So, the two equivalent equations that describe the growth are or Complete the following input/output table for the two functions above t 0 1 2 3 26 2848 31197 34172 26 2848 31197 34174 Notice that the outputs of the two functions are effectively the same
8 (e) If bottled water consumption continues to grow exponentially in the future, what is the projected global consumption in the year 2010? In the year 2010, t = 11 years When t = 11, So, if bottled water consumption is growing exponentially, then the projected global bottled water consumption is approximately71 billion gallons in the year 2010 Example 13: Determine the equations of each of the following exponential functions described below Outputs decay 10% over 4-unit periods and graph has a vertical intercept equal to 7 We begin with the algebraic form The 10% decay over 4-unit periods tells us that c = 090 and n = 4 Since b represents the vertical intercept, b = 7 Thus, the equation is Graph passes through the points (0,1) and (3,4) The two points listed represent input/output pairs for this function: x 0 3 y 1 4 From the table, we determine that the growth factor c equals 4/1 = 4 over 3-unit periods, ie n = 3 Since b represents the value of the output variable corresponding to an input value of 0, b = 1 Thus, the equation is Graph passes through the points (1,10) and (3,4) The two points listed represent input/output pairs for this function: x 1 3 y 10 4 From the table, we determine that the growth factor c equals 4/10 = 04 and n = 2 Thus, the equation is To find b, we will use one of the input/output pairs Since y = 10 when x = 1, it follows that We solve this equation for b Dividing both sides of this equation by which is approximately 1581 Thus, the resulting equation is gives
9 Graph passes through the points (5, 2) and (2, 1) The two points listed represent input/output pairs for this function: x 5 2 y 2 1 From the table, we determine the factor for this function If you think of the values of x as times, then x = 2 is earlier than x = 5 So, the factor c will equal 2/1 = 2 for n = 3 Consequently, the equation is To find b, we again use one of the input/output pairs Let s choose (5, 2) Substituting this pair into our equation yields gives Dividing both sides of this equation by Consequently, the equation is
10 Section 10 Homework Assignment 1 Simplify each of the following expressions using only your knowledge of roots and the exponent properties from class Don t use a calculator! (e) (f) (g) (h) (i) (j) (k) (l) 2 Use your graphing calculator to determine a decimal value (approximation) for each of the following expressions 3 Shown below are some exponential functions Use exponent properties to write the functions in the form What do your results say about the growth/decay factors of each of the functions? 4 Consider the exponential function given by the equation
11 Complete the following input/output table by hand (ie not using a calculator) t 0 1 2 3 4 y Write this function as 5 Determine equations for the exponential functions described below Outputs grow by 20% over 6-unit periods and the graph passes through the point (0,10) Outputs decay by 30% over 10-unit periods and the graph has a vertical intercept equal to 18 The graph passes through the points (0,20) and (4, 15) The graph passes through the points (2,4) and (5, 6) (e) The graph passes through the points (6,10) and (9,9) 6 Determine equations describing the exponentially growing and decaying quantities given below The initial amount is 42 and quantity decays 15% every 5 years The initial amount is 124 and quantity grows by a factor of 112 every 4 months The initial amount is 200 and the quantity doubles every 6 weeks 7 For each of the situations in exercise 6, determine the percentage growth/decay per unit time (eg per year, per month, etc) by using exponent properties 8 This exercise again refers back to exercise 6 Use the equation for the function in part to determine the amount after 20 years Use the equation for the function in part to determine the amount after 1 year Use the equation for the function in part to determine the amount after 26 weeks 9 In this exercise, we look at per capita bottled water consumption in Italy (highest per capita consumer in the world) and the US (highest total consumption in the world) The data is shown in the table below: Bottled water consumption (per capita) in liters 1999 2004 Italy 1548 1836 US 636 905 (Source: Bottled Water 2004: US and International Statistics and Developments, Beverage Marketing Corporation (2005)) Determine the percentage growth in per capita bottled water consumption for the Italy and the US during this period If we assume that consumption in both countries grew by constant annual percentages during this period, determine these percentages Determine the average rate of change in per capita consumption for both countries during this period Be sure you give units for your answer
12 Estimate per capita consumption in both countries for each year during this period, assuming that consumption grows by a constant annual percentage Year 1999 2000 2001 2002 2003 2004 Bottled water consumption Italy 1548 1836 (per capita) in liters US 636 905 (e) Estimate per capita consumption in both countries for each year during this period, assuming that consumption grows at a constant rate of change Year 1999 2000 2001 2002 2003 2004 Bottled water consumption Italy 1548 1836 (per capita) in liters US 636 905 10 Input/output tables for one linear function and one exponential function are given below For the exponential function, determine the constant growth/decay factor over 1-unit periods For the linear function, determine the constant rate of change Then, determine equations for these two functions x 1 3 6 y 24 0864 018662 x 1 3 6 y 725 975 135
13 Section10 Answers to Selected Homework Exercises 1 (f) (g) (h) (i) 2 3 5 Because, growing by a factor of 16 during 2-unit periods of time is equivalent to growing by a factor of 4 over 1-unit periods of time y = 80( 27) t 3 = 80( 3) t Because, growing by a factor of 27 during 3-unit periods of time is equivalent to growing by a factor of 3 over 1-unit periods of time 6 7 29% monthly 12% weekly 8 174211 403175 9 Italy 186%; US 423% Italy 35%; US 73%
14 Italy 576 liters/year; US 538 liters/year Year 1999 2000 2001 2002 2003 2004 Bottled water consumption Italy 1548 1601 1657 1715 1774 1836 (per capita) in liters US 636 682 732 786 843 905 (e) Year 1999 2000 2001 2002 2003 2004 Bottled water consumption Italy 1548 1606 16632 1721 1778 1836 (per capita) in liters US 636 690 744 797 851 905