Exponential Growth and Decay Models Exploration
Examine various exponential functions in real-life scenarios, understanding their shapes, properties, and asymptotes. Learn to identify key parameters like y-intercept, growth rate, and asymptotic lines.
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8 March, 2025 Exponential growth and decay models LO: Use exponential functions to model real-life situations. www.mathssupport.org
Exponential function An exponential function has a number, called the base, raised to a power, In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 Let s look at the shape of some exponential functions y = 2x (0, 1) the independent variable (x) is the index or power. What is the y-intercept? y Is the graph increasing or decreasing? Increasing. It is called exponential Growth What is the value of k? of a? 2 1 of c? 0 What is the line that the graph get closer to, but never touch? The x-axis. Its equation isy = 0 x It is called an asymptote www.mathssupport.org
Exponential function An exponential function has a number, called the base, raised to a power, In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 Let s look at the shape of some exponential functions y = 3x (0, 1) the independent variable (x) is the index or power. What is the y-intercept? y Is the graph increasing or decreasing? Increasing. It is called exponential Growth What is the value of k? of a? 3 1 of c? 0 What is the line that the graph get closer to, but never touch? The x-axis. Its equation isy = 0 x It is called an asymptote www.mathssupport.org
Exponential function An exponential function has a number, called the base, raised to a power, In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 Let s look at the shape of some exponential functions y = 3(2)x (0, 3) the independent variable (x) is the index or power. What is the y-intercept? y Is the graph increasing or decreasing? Increasing. It is called exponential Growth What is the value of k? of a? 2 3 of c? 0 What is the line that the graph get closer to, but never touch? The x-axis. Its equation isy = 0 x It is called an asymptote www.mathssupport.org
Exponential function An exponential function has a number, called the base, raised to a power, In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 Let s look at the shape of some exponential functions y = 3(2)x - 1 (0, 3) the independent variable (x) is the index or power. What is the y-intercept? y Is the graph increasing or decreasing? Increasing. It is called exponential Growth What is the value of k? of a? 2 3 of c? -1 What is the line that the graph get closer to, but never touch? The line with equationy = 1 It is called an asymptote x www.mathssupport.org
Exponential function An exponential function has a number, called the base, raised to a power, In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 Let s look at the shape of some exponential functions What is the y-intercept? (0, 1) ? the independent variable (x) is the index or power. ? ? y = y Is the graph increasing or decreasing? Decreasing. It is called exponential Decay What is the value of k? of a? 2 1 of c? 0 1 What is the line that the graph get closer to, but never touch? The x-axis. Its equation isy = 0 x It is called an asymptote www.mathssupport.org
Exponential function An exponential function has a number, called the base, raised to a power, In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 Let s look at the shape of some exponential functions What is the y-intercept? (0, 4) ? the independent variable (x) is the index or power. ?+ 3 ? y = y Is the graph increasing or decreasing? Decreasing. It is called exponential Decay What is the value of k? of a? 2 1 of c? 3 1 What is the line that the graph get closer to, but never touch? The line with equationy = 3 It is called an asymptote x www.mathssupport.org
Exponential function Sumarising. In general, the form of an exponential function is: f(x) = kax + c where a > 0, a 1andk 0 The parameter k is the vertical stretch factor If a > 1 the graph is increasing so, it is exponential growth If 0 < a < 1 the graph is decreasing so, it is exponential decay The parameter c is the vertical translation of the graph. The equation of the horizontal asymptote is y = c The y-intercept is the point (0, k + c) www.mathssupport.org
Sketching the graph of an exponential function You need to be careful when copying the graph from your GDC onto paper. f(x) = 4(25)0.18x+ 2 3 x 3 Sketch the graph of Draw your axes and label them Make sure that any x- and y-intercepts are in the correct place. Determine the position of any asymptote. Plot the starting and ending points based on the given domain y (3, 24.75) 25 (0, 6) ( 3, 2.70) y = 2 x 0 3 3 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx Turn on the calculator. 1990 4 978 496 Click STAT (Statistics) Year Population 1980 3 717 165 1985 4 278 501 1995 5 905 558 2000 6 865 951 2005 7 982 225 2010 9 199 259 2015 10 575 952 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx 1985 4 278 501 Turn on the calculator. Click STAT (Statistics) Press 1 (Edit) Year Population 1980 3 717 165 1990 4 978 496 1995 5 905 558 2000 6 865 951 2005 7 982 225 2010 9 199 259 2015 10 575 952 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx 1985 4 278 501 (85) Turn on the calculator. Click STAT (Statistics) Press 1 (Edit) Year Population (80) 1980 3 717 165 (90) 1990 4 978 496 (95) 1995 5 905 558 Clear all lists in the GDC Type the data for Year in List 1 and Population in List 2 To make it simpler we will change the year numbers (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx Turn on the calculator. 1990 4 978 496 (90) Click STAT (Statistics) Press 1 (Edit) Year Population (80) 1980 3 717 165 (85) 1985 4 278 501 (95) 1995 5 905 558 Clear all lists in the GDC Type the data for Year in List 1 and Population in List 2 Press 2nd statplot (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx Turn on the calculator. 1990 4 978 496 (90) Click STAT (Statistics) Press 1 (Edit) Year Population (80) 1980 3 717 165 (85) 1985 4 278 501 (95) 1995 5 905 558 Clear all lists in the GDC Type the data for Year in List 1 and Population in List 2 Press 2nd statplot Press 1 (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx Turn on the calculator. 1990 4 978 496 (90) Click STAT (Statistics) Press 1 (Edit) Year Population (80) 1980 3 717 165 (85) 1985 4 278 501 (95) 1995 5 905 558 Clear all lists in the GDC Type the data for Year in List 1 and Population in List 2 Press 2nd statplot Press 1 Plot1 ON Xlist: L1 Ylist: L2 ZOOM 9 ZoomStat (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx Turn on the calculator. 1990 4 978 496 (90) Click STAT (Statistics) Press 1 (Edit) Year Population (80) 1980 3 717 165 (85) 1985 4 278 501 (95) 1995 5 905 558 Clear all lists in the GDC Type the data for Year in List 1 and Population in List 2 Press 2nd statplot Press 1 Plot1 ON Xlist: L1 Ylist: L2 ZOOM 9 ZoomStat Press MODE. (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx 1985 4 278 501 (85) Use the arrow to highlight STAT DIAGNOSIS ON Press STAT Year Population (80) 1980 3 717 165 (90) 1990 4 978 496 (95) 1995 5 905 558 (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx 1985 4 278 501 (85) Use the arrow to highlight STAT DIAGNOSIS ON Press STAT Scroll down 0:ExpReg Year Population (80) 1980 3 717 165 (90) 1990 4 978 496 CALC (95) 1995 5 905 558 (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Find an exponential function that models the data of the form y = abx 1985 4 278 501 (85) Use the arrow to highlight STAT DIAGNOSIS ON Press STAT Scroll down 0:ExpReg Xlist: L1 Ylist: L2 Store RegEqPress Vars Y-VARS 1-Function Year Population (80) 1980 3 717 165 (90) 1990 4 978 496 CALC (95) 1995 5 905 558 (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 1-Y1 2010 9 199 259 Scroll down Calculate ENTER (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Year The general exponential function is: Population (80) 1980 3 717 165 y = abx (85) 1985 4 278 501 From the GDC we get a = 338 955 The data can be modelled with the function (90) 1990 4 978 496 b = 1.03 (95) 1995 5 905 558 (100) 2000 6 865 951 y = 338 955(1.03)x The coefficient of determination R2 reveals that 99.9% of the data fit the regression model Drawing the graph (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 ZOOM 9 ZoomStat www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Year The general exponential function is: Population (80) 1980 3 717 165 y = abx (85) 1985 4 278 501 From the GDC we get a = 338955 The data can be modelled with the function (90) 1990 4 978 496 b = 1.03 (95) 1995 5 905 558 (100) 2000 6 865 951 y = 338 955(1.03)x The coefficient of determination R2 reveals that 99.9% of the data fit the regression model Drawing the graph (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 ZOOM 9 ZoomStat www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Year Estimate the population of Benin in 1997. CALC Population (80) 1980 3 717 165 (85) 1985 4 278 501 1:ValueType 97 2nd TRACE (90) 1990 4 978 496 (95) 1995 5 905 558 (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Year Estimate the population of Benin in 1997. CALC Population (80) 1980 3 717 165 (85) 1985 4 278 501 1:ValueType 97 2nd TRACE ENTER (90) 1990 4 978 496 (95) 1995 5 905 558 (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
Modelling with a exponential function Exponential functions can be used to model real world data. The data in the following table list the population of Benin from 1980 through 2019. (Source: https://data.worldbank.org/) Year Estimate the population of Benin in 1997. CALC Population (80) 1980 3 717 165 (85) 1985 4 278 501 1:ValueType 97 2nd TRACE ENTER (90) 1990 4 978 496 (95) 1995 5 905 558 The population of Benin in 1997 was approximately 6 198 189. (100) 2000 6 865 951 (105) 2005 7 982 225 (110) 2010 9 199 259 (115) 2015 10 575 952 (119) 2019 11 801 151 www.mathssupport.org
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