8.5 Exponential Growth and 8.6 Exponential Decay FUNctions

What am I going to learn? Concept of an exponential growth and decay function Models for exponential growth/decay

New Concepts Independent variable is another name for domain or input, which is typically but not always represented using the variable, x. Dependent variable(function value) is another name for range or output, which is typically but not always represented using the variable, y.

What is an exponential growth/decay FUNction? Exponential Growth FUNction – The function value (y) is increasing exponentially if it increases by the same rate in each unit of input (x). Exponential Decay FUNction – The function value (y) is decreasing exponentially if it decreases by the same rate in each unit of input (x).

What is a quantity increases exponentially for each unit of the input? Initial Layer 1 x 3 1st Layer 3 x 3 2nd Layer 9 x 3 3rd Layer 27

What is a quantity decreases exponentially for each unit of the input? Initial Square Area = 322 4 x 1/4 1st Square Area = 162 8 x 1/4 2nd Square Area = 82 16 x 1/4 3rd Square Area = 42 x 1/4 4th Square Area = 22 32

What is an exponential function? Obviously, it must have something to do with an exponent! An exponential function is a function whose independent variable is an exponent.

Group Activity 1 From the tree structure example, if the input (independent) variable x = the layer number, and the output (dependent) variable y = the number of nodes, student A, B, and C wrote 3 equations that best described the information in the slide. x 1 2 3 … y 9 27 Student A wrote Student B wrote Student C wrote Identify which student is correct. How do you know? Then explain why the other students are not correct.

Student A is not correct Student A is not correct. Although he/she related y and x exponentially, the number of nodes in the current layer is not related the layer number correctly. Student C is not correct. He/She wrote an equation that related y and x linearly instead of an exponential one. This student may just noticed the pattern was time 3 and may not have known what to do with it. Student B is correct. He/She knows the quantity (y) increases 3 times as the layer number (x) is increased by 1.

Group Activity 2 From the square area example, if the input (independent) variable x = the time of division, and the output (dependent) variable y = area of colored square, student A, B, and C wrote 3 equations that best described the information in the slide. x 1 2 3 4 … y 162 82 42 22 Student A wrote Student B wrote Student C wrote Identify which student is correct. How do you know? Then explain why the other students are not correct.

Student A is not correct Student A is not correct. Although he/she related y and x exponentially, the area in each division is not related to the number of times of division correctly. Student B is not correct. He/She wrote an equation that related y and x exponentially, however, he/she did not related the quantity of area to the number of times of division correctly. This student may just noticed the pattern was time ¼ and may not have known what to do with it correctly. Student C is correct. He/She knows the quantity of the area (y) decreases ¼ times as the number of times of division (x) is increased by 1. He/She related the quantities y and x correctly.

What does an exponential function look like? y=ab Exponent, the Independent Variable Value, Dependent Variable Base such as 3, or ¼ Just some number that’s not 0 Why not 0?

The Basis of Bases The base of an exponential function carries much of the meaning of the function. The base determines exponential growth or decay. The base is a positive number; however, it cannot be 1.

Exponential Growth/Decay An exponential function models growth whenever its base > 1. (Why?) If the base b > 1, then b is referred to as the growth factor. An exponential function models decay whenever its base < 1. (Why?) If the base b < 1, then b is referred to as the decay factor.

EXPONENTIAL GROWTH MODEL In many cases, quantity does not increases by the integer number of times, instead, the quantity increases by the some rate r in each time period t. When this happens, the value of the quantity at any given time can be calculated as a function of the rate and the initial amount. EXPONENTIAL GROWTH MODEL C is the initial amount. t is the time period. y = C (1 + r)t (1 + r) is the growth factor, r is the growth rate.

Exponential Growth Functions If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the equation Where C = initial amount r = growth rate (percent written as a decimal) t = time where t  0 (1 + r) = growth factor where 1 + r > 1

EXPONENTIAL DECAY MODEL In many cases, quantity does not decreases by ½, 1/3, ¼, etc, instead, the quantity decreases by the some rate r in each time period t. When this happens, the value of the quantity at any given time can be calculated as a function of the rate and the initial amount. EXPONENTIAL DECAY MODEL C is the initial amount. t is the time period. y = C (1 – r)t (1 – r ) is the decay factor, r is the decay rate.

Exponential Decay Functions If a quantity decreases by the same proportion r in each unit of time, then the quantity displays exponential decay and can be modeled by the equation Where C = initial amount r = growth rate (percent written as a decimal) t = time where t  0 (1 – r) = decay factor where 1 – r < 1

EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL C is the initial amount. t is the time period. y = C (1 + r)t (1 + r) is the growth factor, r is the growth rate. EXPONENTIAL DECAY MODEL C is the initial amount. t is the time period. y = C (1 – r)t (1 – r ) is the decay factor, r is the decay rate.

Example 1: Exponential Growth The original value of a painting is $9,000 and the value increases by 7% each year. Write an exponential growth function to model this situation. Then find the painting’s value in 15 years. Step 1 Write the exponential growth function for this situation. y = C(1 + r)t Write the formula. Substitute 9000 for C and 0.07 for r. = 9000(1 + 0.07)t = 9000(1.07)t Simplify.

Example 1: Exponential Growth The original value of a painting is $9,000 and the value increases by 7% each year. Write an exponential growth function to model this situation. Then find the painting’s value in 15 years. Step 2 Find the value in 15 years. y = 9000(1.07)t Call the model in last step. = 9000(1 + 0.07)15 Substitute 15 for t. Use a calculator and round to the nearest hundredth. ≈ 24,831.28

Check Up Example 1 A sculpture is increasing in value at a rate of 8% per year, and its value in 2000 was $1200. Write an exponential growth function to model this situation. Then find the sculpture’s value in 2006. Step 1 Write the exponential growth function for this situation. y = C(1 + r)t Write the formula Substitute 1200 for C and 0.08 for r. = 1200(1 + 0.08)t = 1200(1.08)t Simplify.

Check Up Example 1 A sculpture is increasing in value at a rate of 8% per year, and its value in 2000 was $1200. Write an exponential growth function to model this situation. Then find the sculpture’s value in 2006. Step 2 Find the value in 6 years. y = 1200(1.08)t = 1200(1 + 0.08)6 Substitute 6 for t. Use a calculator and round to the nearest hundredth. ≈ 1,904.25 The value of the sculpture in 6 years is $1,904.25.

Example 2 Exponential Growth Moore’s law, a rule used in the computer industry, states that the number of transistors per integrated circuit (the processing power) doubles every year. Beginning in the early days of integrated circuits, the growth in capacity is expressed by the following table. Write the exponential growth model and tell the growth rate. Then predict the capacity in 1998. Note: Double means growth factor = 2, 1965 as initial year y = 60(2)t 1 + r = 2, r = 1

Example 2 Exponential Growth Moore’s law, a rule used in the computer industry, states that the number of transistors per integrated circuit (the processing power) doubles every year. Beginning in the early days of integrated circuits, the growth in capacity is expressed by the following table. Write the exponential growth model and tell the growth rate. Then predict the capacity in 1998. 1998 is 33 years since 1965 y = 60(2)33 y = 515,396,075,520

Example 3: Exponential Decay The population of a town is decreasing at a rate of 3% per year. In 2000 there were 1700 people. Write an exponential decay function to model this situation. Then find the population in 2011. Step 1 Write the exponential decay function for this situation. y = C(1 – r)t Write the formula. Substitute 1700 for C and 0.03 for r. = 1700(1 – 0.03)t = 1700(0.97)t Simplify.

Example 3: Exponential Decay The population of a town is decreasing at a rate of 3% per year. In 2000 there were 1700 people. Write an exponential decay function to model this situation. Then find the population in 2011. Step 2 Find the population in 2011. y = 1,700(0.97)11 Substitute 11 for t. Use a calculator and round up to the whole number which is greater than the decimal. ≈ 1216.01 ≈ 1217

Check Up Example 3 You buy a new car for $22,500. The car depreciates at the rate of 7% per year, What was the initial amount invested? What is the decay rate? The decay factor? What will the car be worth after the first year? The second year? The initial investment was $22,500. The decay rate r = 0.07. The decay factor (1 – r) = 0.93.

Example: Compound Interest You deposit $1500 in an account that pays 2.3% interest compounded yearly, What was the initial principal (P) invested? What is the growth rate (r)? The growth factor? Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money? The initial principal (P) is $1500. The growth rate (r) is 0.023. The growth factor is 1.023.

Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.

There will be about 4860 rabbits after 5 years. Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. b. What is the population after 5 years? Help SOLUTION “population triples” means (1 + r) = 3, or, r = 2 After 5 years, the population is P = C(1 + r) t Exponential growth model = 20(1 + 2) 5 Substitute C, r, and t. = 20 • 3 5 Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years.

GRAPHING EXPONENTIAL GROWTH MODELS A Model with a Large Growth Factor Graph the growth of the rabbit population. SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t P 4860 60 180 540 1620 20 5 1 2 3 4 1000 2000 3000 4000 5000 6000 1 7 2 3 4 5 6 Time (years) Population Here, the large growth factor of 3 corresponds to a rapid increase P = 20 ( 3 ) t

The purchasing power of a dollar in 1997 compared to 1982 was $0.59. Writing an Exponential Decay Model COMPOUND INTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? SOLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. y = C (1 – r) t Exponential decay model = (1)(1 – 0.035) t Substitute 1 for C, 0.035 for r. = 0.965 t Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = 0.96515 Substitute 15 for t. 0.59 The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

Purchasing Power (dollars) GRAPHING EXPONENTIAL DECAY MODELS Graphing the Decay of Purchasing Power Help Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t y 0.837 0.965 0.931 0.899 0.867 1.00 5 1 2 3 4 0.7 0.808 0.779 0.752 0.726 10 6 7 8 9 0.2 0.4 0.6 0.8 1.0 1 12 3 5 7 9 11 Years From Now Purchasing Power (dollars) 2 4 6 8 10 y = 0.965t Your dollar of today will be worth about 70 cents in ten years.

Make a table of values for the function Your Group Try It Make a table of values for the function using x-values of –2, -1, 0, 1, and Graph the function. Does this function represent exponential growth or exponential decay?

Problem 1 This function represents exponential decay.

Your GroupTry It Your business had a profit of $25,000 in 1998. If the profit increased by a factor of 1.12 each year, to 108% each year, what would your expected profit be in the year 2010? Identify C, t, r, and the growth factor. Write down the equation you would use and solve.

Problem 2 a) C = $25,000 t = 12 r = 0.12 Growth factor = 1.12

Problem 2 b) C = $25,000 t = 12 r = 108% – 100% = 8% = 0.08 Growth factor = 1.08

You Try It 3) Iodine-131 is a radioactive isotope used in medicine. Its decay rate is 36.5%. If a patient is given certain amount mg of iodine-131, it is about 0.66 mg left after 8 days, what was the initial amount of dosage? Identify C, t, r, and the decay factor. Write down the equation you would use and solve.

Problem 3 C = x mg t = 8 r = 0.365 Decay factor = 1 – 0.365 = 0.635

Example 4 Identify each of the following as an exponential growth or decay function; tell the growth or decay rate; tell the growth or decay factor. 1. f (x) = 0.324(1+0.03)x 2. f (x) = 279(1 – 0.05)x 3. f (x) = 100(1 – 0.12)x 4. f (x) = 0.6977(1 + 0.08)x Exp. Growth, 0.03, 1.03 Exp. Decay, 0.05, 0.95 Exp. Decay, 0.12, 0.88 Exp. Growth, 0.08, 1.08

Investigation: Tournament Play The NCAA holds an annual basketball tournament every March. The top 64 teams in Division I are invited to play each spring. When a team loses, it is out of the tournament. Work with a partner close by to you and answer the following questions.

Investigation: Tournament Play Fill in the following chart and then graph the results on a piece of graph paper. Then be prepared to interpret what is happening in the graph. After round x Number of teams in tournament (y) 64 1 32 2 16 3 8 4 5 6 After round x Number of teams in tournament (y) 64 1 2 3 4 5 6

Asymptotes Notice that the left side of the graph gets really close to y = 0 as . We call the line y = 0 an asymptote of the graph. Think about why the curve will never take on a value of zero and will never be negative.

Cool Fact: All exponential growth functions look like this! Asymptotes Cool Fact: All exponential growth functions look like this!

Asymptotes Notice the right side of the graph gets really close to y = 0 as . We call the line y = 0 an asymptote of the graph. Think about why the graph will never take on a value of zero and will never be negative.

What does Exponential Decay look like? Graph: Cool Fact: All exponential decay functions look like this! Table of Values: x (½)x y -2 ½-2 4 -1 ½-1 2 ½0 1 ½1 ½ ½2 ¼ 3 ½3 1/8

EXPONENTIAL GROWTH AND DECAY MODELS GRAPHING EXPONENTIAL DECAY MODELS EXPONENTIAL GROWTH AND DECAY MODELS CONCEPT SUMMARY EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL y = C (1 + r)t y = C (1 – r)t An exponential model y = a • b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1. (0, C) (1 – r) is the decay factor, r is the decay rate. (1 + r) is the growth factor, r is the growth rate. C is the initial amount. t is the time period. 1 + r > 1 0 < 1 – r < 1

Match the following function(s) to the appropriate graph. (More than one function can match the same graph) U V W Graph U: B, C, J Graph V: E, K Graph W: G, H, I

Challenge Is the product of two exponential growth functions still exponential growth? Is the product of two exponential decay functions still exponential decay? What is the ratio of an exponential growth functions to an exponential decay function? What is the ratio of an exponential decay functions to an exponential growth function? Hint: For question 1 and 2, give yourself two same type of functions. For question 3 and 4, give yourself two different type of functions.

Challenge Is the product of two exponential growth functions still exponential growth? [Answer] Yes. Example y1 = 3(1.08)x, y2 = 2(1.32)x Then y1 y2= 3(1.08)x  2(1.32)x = 6(1.08 1.32)x = 6(1.4256)x Rearrange by using the “Power of Product” property Simply in one word, the product of two positive real numbers both greater than 1 is still greater than 1

Challenge 2) Is the product of two exponential decay functions still exponential decay? [Answer] Yes. Example y1 = 2(0.78)x, y2 = 5(0.32)x Then y1 y2= 2(0.78)x  5(0.32)x = 10(0.78 0.32)x = 10(0.2496)x Rearrange by using the “Power of Product” property Simply in one word, the product of two positive real numbers both less than 1 is still less than 1.

Challenge 3) What is the ratio of an exponential growth functions to an exponential decay function? [Answer] Yes. Example y1 = 4(1.12)x, y2 = 8(0.16)x Then y1 y2= 4(1.12)x  8(0.16)x = (4 8)(1.12 0.16)x = 0.5(7)x Rearrange by using the “Power of Quotient” property Simply in one word, the ratio of one positive real numbers which is greater than 1 to one positive real number which less than 1 is greater than 1.

Challenge 4) What is the ratio of an exponential decay functions to an exponential growth function? [Answer] Yes. Example y1 = 3(0.48)x, y2 = 12(3)x Then y1 y2= 3(0.48)x  12(3)x = (312)(0.48  3)x = 0.25(0.16)x Rearrange by using the “Power of Quotient” property Simply in one word, the ratio of one positive real numbers which is less than 1 to one positive real number which greater than 1 is less than 1.

Extremely Challenge 5) Is the product of one exponential growth functions and one exponential decay function still exponential growth? 6) Is the ratio of two exponential decay functions still exponential decay? Hint: For question 5 and 6, give yourself two sets of two same type of functions. Answers to both questions are the same: “Not certain, it depends on the given functions”