VIEWS: 10 PAGES: 17 POSTED ON: 7/26/2011
Teacher Notes Exponential Functions : Exponential Growth & Decay Topic: This lesson introduces students to the concept of exponential growth and decay. Objective: Given a geometric pattern the student will derive an exponential equation by analyzing data for common ratios. Time Table: This lesson will need two 45 minute periods or one 90 minute period. TEKS Focus: A.11C Analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods. Materials: Lesson Packets (provided) Doubling sheets (provided) Halving sheets (provided) Letter sized paper Colored pencils to color doubling and halving sheets (optional) Graphing calculators T I Interactive and cable (optional) Printer (optional) Lesson Overview: There are 4 major activities : Doubling Method, Halving Method, Folding paper and Page Area. Students use the doubling sheet on Page 5 to derive an exponential growth function (y = 2x) 1 and the halving sheet on Page 6 to derive an exponential decay function (y = ( ) x ). More exponential 2 growth and decay functions can be derived from the folding paper and page area activities. A general exponential representation y = abx can be derived from the activities. Students summarize and describe each parameter from their understanding. Graphic representations of exponential growth and decay are 1 compared using the previously derived functions, y = 2x and y = ( ) x . 2 Grouping: The doubling activity should be led by the teacher and the students can work the rest of the lesson in groups. Procedures: Teacher clarifies the objectives of the lesson and uses them as part of the informal assessment (check for understanding) during the lesson. Teacher leads off by coloring squares on the doubling sheet and filling in the doubling table. Students should be developing an exponential growth pattern and able to derive its equation. As a class, discuss the Zero Power on Page 1. If students feel comfortable with the doubling activity, let students work in groups on the halving, folding paper and page area activities. As a class, discuss the summary on page 3 by asking students to brainstorm possible 1 descriptions of each parameter: y, x, a and b. Have students graph y = 2x and y = ( ) x , sketch or 2 capture their graphs using TI Interactive and paste/staple them on the provided space on Page 4. Compare and discuss the characteristics of both graphs. Practice problems can be answered as a class or in groups. Homework: Interating on a Plane. This is an integration of geometry and exponential functions. Students build patterns for numbers of squares and surface areas after iterations. They are to derive exponential functions and graph the functions. MMA Name: _____KEY___________ Date: _________ Period:______ Objectives: To derive exponential functions using the doubling and halving methods. To differentiate between exponential growth and decay by comparing their graphs and algebraic expressions. Use the doubling sheet on page 5 to fill in the doubling table. Number of Number of Squares Growth Exponent Form Doubles (d) (n) Factor 0 1 20 - 1 2 21 2 2 4 22 2 3 8 23 2 4 16 24 2 5 32 25 2 d 2d 2 The number of squares (n) is a function of the number of doublings (d). n = ……… 2d …….…. or y = ……… 2x ………… or f(x) = ……… 2x ……….. When the independent variable (x) is the exponent, the function is an ‘exponential function’. The doubling table shows an exponential growth. Number of Doubles Number of Squares Exponent Form 0 1 20 Any number to the ………………… equals ……….. You are an heir to ($1,000,000)0. How much will you inherit? …………$1…………… Use the halving sheet on page 6 to fill in the halving table. Number of Number of Squares Decay Exponent Form Halves (h) (n) Factor 0 1 ( 1 )0 - 2 1 1 2 ( 1 )1 1 2 2 2 1 4 ( 1 )2 1 2 2 3 1 8 ( 1 )3 1 2 2 4 1 16 ( 1 )4 1 2 2 5 1 32 ( 1 )5 1 2 2 h ( 1 )h 1 2 2 The number of squares (n) is a function of the number of halving (h). h h h 1 1 1 n = …… ….…. or y = ……… ……… or f(x) = …… …… 2 2 2 When the independent variable (x) is the exponent, the function is an ‘exponential function’. The halving table shows an exponential decay. Use a piece of letter sized paper for this activity. Number of Number of Leaves Number of Pages Exponent Form folds (f) (L) (p) 0 1 2 2 20 1 2 4 2 21 2 4 8 2 22 3 8 16 2 23 4 16 32 2 24 5 32 64 2 25 f 2 2f number of leaves, L = 2 f f f+1 x+1 number of pages, p = 2(2 ) = 2 OR y = 2 Let’s start with a paper size of 8 inch wide and 10 inch long. Number of Page Area (A) Exponent Page Width (in) Page Length (in) folds (f) (in2) Form 0 8 10 80 80( 1 2 )0 1 4 10 40 80( 1 )1 2 2 2 10 20 80( 1 ) 2 2 3 1 10 10 80( 1 )3 2 4 0.5 10 5 80( 1 ) 4 2 5 0.25 10 2.5 80( 1 )5 2 f 80( 1 )f 2 1 f 1 x Page Area, A = 88( ) or y = …88( )…………….. 2 2 - independent variable - domain - dependent variable - input - range - exponent - amount after x events - number of events - coefficient - base - original amount - proportional growth if b > 1 - starting amount - decay factor if b < 1 Exponential Growth: b is (less than or greater than) 1. Exponential Decay: b is (less than or greater than) 1. Graph y 2x and use TI-Interactive to capture your screen. Print your graph and paste it 1 in the space provided below. Repeat with y ( ) x . 2 Graph of exponential growth Graph of exponential decay If do not have TI Interactive, have student hand sketch the graphs. PRACTICE PROBLEMS Evaluate and predict whether each of the following functions is an exponential growth or decay. 1 a. y 3x when x = 4 b. y ( ) x when x = 4 3 y = …81 ……. y = …… 0.012…. growth vs. decay ……… growth …. growth vs. decay ……decay……. c. f (x) 2x 1 when x = 3 d. f (x) 2 x 1 when x = 3 y = ……9…………. y = ……16…………. growth vs. decay …… growth ……. growth vs. decay …… growth …. e. P ( 2 )q 1 when q = 2 f. f (x) ( 2 )x 1 when x = 2 3 3 y = …1.44…………. y = ………0.296………. growth vs. decay …… decay ……. growth vs. decay … decay ………. Doubling Sheet (This answer shows one of the many ways of successive doublings.) No. of Doublings No. of Squares 32 5 16 4 8 3 4 2 1 2 1 Halving Sheet (This answer shows one of the many ways of successive halvings.) 0th halving 1st halving 2nd halving 3rd halving 4th halving 5th halving MMA Name: Solution Exp Growth and Decay Period: ___ Date: ________ Exponential Growth and Decay – Iterating on a Plane 1. If the area of the original square (stage 0) is 9 units2, what is the area of the new figure at stage 1? 5 2. The area of the stage 1 figure is what fraction of the area of the stage 0 figure? 5/9 3. When the figure goes from stage 0 to stage 1, does the area increase, decrease, or stay the same? ______________________ 4. What is the perimeter of the stage 0 square? 12 units 5. What is the perimeter of the stage 1 figure? 20 units 6. When the figure goes from stage 0 to stage 1, does the perimeter increase, decrease, or stay the same? ______________________ An iteration rule has three distinct parts: a reduction, a replication, and a rebuilding. Each time the rule is applied, the result is a new and more complex figure. Fill in the tables. Graph your results. Remember to label your axes and scale. Number of Squares Number of Stage Process Squares 0 50 1 5 51 5 1 2 5(5) 52 25 3 5(5)(5) 53 125 4 5(5)(5)(5) 54 625 n 5n 5n From stage to stage, the number of squares increases/decreases by a factor of 5. Total area (units2) Stage Process Total area 5 0 9( )0 9 9 5 5 1 9( ) 9( )1 5 9 9 5 5 5 2 25 2 9( )( ) 9( ) 9 9 9 9 5 5 5 5 125 9( )( )( ) 9( )3 3 9 9 9 9 81 5 5 5 5 5 625 4 9( )( )( )( ) 9( )4 9 9 9 9 9 819 5 n 9( )n 9 From stage to stage, the total area increases/decreases by a factor of 5/9 . MMA Name: ________________ Date: _________ Period:______ Objectives: To derive exponential functions using the doubling and halving methods. To differentiate between exponential growth and decay by comparing their graphs and algebraic expressions. Use the doubling sheet on page 5 to fill in the doubling table. Number of Number of Squares Growth Exponent Form Doubles (d) (n) Factor 0 1 - d The number of squares (n) is a function of the number of doublings (d). Which can be written using different notation but are the same n= or y= or f(x) = When the independent variable (x) is the exponent, the function is an ‘exponential function’. The doubling table shows an exponential growth. Look at the first row in the table above. Number of Doubles Number of Squares Exponent Form 0 Any number to the power equals You are an heir to ($1,000,000)0. How much will you inherit? $ Use the halving sheet on page 6 to fill in the halving table. Number of Number of Squares Decay Exponent Form Halves (h) (n) Factor 0 1 - h The number of squares (n) is a function of the number of halving (h). n= or y= or f(x) = When the independent variable (x) is the exponent, the function is an ‘exponential function’. The halving table shows an exponential decay. Use a piece of letter sized paper for this activity. Number of Number of Leaves Number of Pages Exponent Form folds (f) (L) (p) 0 1 2 f number of leaves, L = 2 number of pages, p = 2(2 ) = 2 or y=2 Let’s start with a paper size of 8 inch wide and 10 inch long. Each time you fold only fold the width (shorter side) in half. Number of Page Area (A) Exponent Page Width (in) Page Length (in) folds (f) (in2) Form 0 8 10 80 80( 1 2 )0 f 1 Page Area, A = 80( ) or y = ……………………….. 2 Exponential Growth: b is (less than or greater than) 1. Exponential Decay: b is (less than or greater than) 1. Graph y 2x and use TI-Interactive to capture your screen (or sketch below). Print your 1 graph and paste it in the space provided below. Repeat with y ( ) x . 2 Graph of exponential growth Graph of exponential decay PRACTICE PROBLEMS Evaluate and predict whether each of the following functions is an exponential growth or decay. 1 a. y 3x when x = 4 b. y ( ) x when x = 4 3 y = ………………. y = ………………. growth vs. decay ………………. growth vs. decay ………………. c. f (x) 2x 1 when x = 3 d. f (x) 2 x 1 when x = 3 y = ………………. y = ………………. growth vs. decay ………………. growth vs. decay ………………. e. P ( 2 )q 1 when q = 2 f. f (x) ( 2 )x 1 when x = 2 3 3 y = ………………. y = ………………. growth vs. decay ………………. growth vs. decay ………………. Doubling Sheet No. of Doublings Draw the Number of Squares 5 4 3 2 1 0 Shade twice the number of squares each time. Record data in table. Halving Sheet Number of Squares (n) = 1 0th halving 1st halving – shade half of the original square 2nd halving – shade half of 3rd halving the previous square 4th halving 5th halving MMA Name: ________________ Date: _________ Period:______ Iterating on a Plane 3 units 3 units 1. If the area of the original square (stage 0) is 9 units2, what is the area of the new figure at stage 1? ______ 2. The area of the stage 1 figure is what fraction of the area of the stage 0 figure? _____ 3. When the figure goes from stage 0 to stage 1, does the area increase, decrease, or stay the same? ______________________ 4. What is the perimeter of the stage 0 square? _____ 5. What is the perimeter of the stage 1 figure? _____ 6. When the figure goes from stage 0 to stage 1, does the perimeter increase, decrease, or stay the same? ______________________ An iteration rule has three distinct parts: a reduction, a replication, and a rebuilding. Each time the rule is applied, the result is a new and more complex figure. 3 units 3 units Fill in the tables. Graph your results. Remember to label your axes and scale. Number of Squares Number of Stage Process Squares 0 1 2 3 4 n From stage to stage, the number of squares increases/decreases by a __________ of ___ . Equation: y = ______________ or f(x) = ______________ Total area (in2) Stage Process Total area 0 1 2 3 4 n From stage to stage, the total area increases/decreases by a __________ of ___ . Equation: y = ______________ or f(x) = ______________