Concept Category 2. Exponential and Log Functions
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1 Concept Category 2 Exponential and Log Functions
2 Concept Category 2 Check List *Find the inverse and composition of functions *Identify an exponential from a table, graph and equation *Identify the difference between a exponential growth and decay in equation and graph forms *Use transformations to graph exponential growth and decay functions. *Create and solve growth and decay word problems. *Solve interest and compound interest problems. *Graph a logarithmic function based on its relationship to an exponential function. *Perform transformations on logarithmic function. *Rewrite exponential equations in the form of a logarithm. *Use simple logarithm rules to solve problems.
3 Graphing & Writing an Exponential Growth Model A population of 50 rabbits escaped into a wildlife region. The population triples each year for 8 years.
4 (Rate)
5 Writing & Graphing an Exponential Growth Model SOLUTION After 8 years, the population is P = C (3) t Exponential growth model = 50(3) 8 = Substitute C, r, and t. Simplify. = 328,050 There will be about 328,050 rabbits after 8 years.
6 Example1 General Growth Formula 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. b = the original 100% + the growth rate y = ab t = 9000(100% + 7%) t = 9000( ) 15 = 9000(1.07) 15 24, The value of the painting in 15 years is $24,
7 Example 2 General Growth Formula A sculpture is increasing in value at a rate of 8% per year, and its value in 2006 was $1200. Write an exponential growth function to model this situation. Then find the sculpture s value in b = the original 100% + growth rate y = a(100% + 8%) t = 1200( ) 11 = 1200(1.08) 11 $
8 Example 3 General Decay Formula You buy a new car for $22,500. The car depreciates at the rate of 7% per year, after 6 years how much is your car worth? b = the original 100% - decay rate y = a(100% - 7%) t = 22500(1-0.07) 6 = 22500(0.93) 6 $
9 Example 4 Compound Formula Finance Application a) You have $1200 invested at a rate of 2% compounded quarterly (not annually); how much do you have after 7 years? b) How about $1200 invested at a rate of 2% compounded monthly for 7 years.?
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11 Reading Math For compound interest annually means once per year (n = 1). quarterly means 4 times per year (n =4). monthly means 12 times per year (n = 12).
12 Example 4 solution : Finance Application $1200 invested at a rate of 2% compounded quarterly; 7 years. b = original 100% + 2% divided by 4 n = 4 = 1200(1.005) 28 = $
13 b) How about $1200 invested at a rate of 2% compounded monthly for 7 years.? b = original 100% + 2% divided by 12 n = t A P ( ) dollars
14 Exponential Growth Equations Word Problems (applications)
15 8. Iodine-131 is a radioactive isotope used in medicine. Its half-life or decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. 9. Your family business had a profit of $25,000 in If the profit increased by 12% each year, what would your expected profit be in the year 2032?
16 Answer Key
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18 Given Parent Graph f ( x) 2 Sketch, find domain, range, end behavior : gx ( ) w( x) 1 2 x3 x2 x x3 ux ( ) 2 3 s( x) 2 2 x
19 And then sketch,and find domain, range, end behavior : f g ( x) ( x) w ( x) The inverse of the Exponential Functions are called Logarithmic Functions u 1 1 ( x) s ( x)
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21 Do you remember all the steps of transformation? Can you find the Asymptotes?
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28 What should you know by now? *Find the inverse and composition of functions *Identify an exponential from a table, graph and equation *Identify the difference between a exponential growth and decay in equation and graph forms *Use transformations to graph exponential growth and decay functions. *Create and solve growth and decay word problems. *Solve interest and compound interest problems. *Graph a logarithmic function based on its relationship to an exponential function. *Perform transformations on logarithmic function. *Rewrite exponential equations in the form of a logarithm. *Use simple logarithm rules to solve problems.
29 Challenge: How long will it take for the population to reach 10 billion? Given f x x g x x h x x 2 ( ) 3( 2) 4 ( ) 2 5 ( ) 2 3 DOK1 a] ( g f )( 2) b] ( f g)( x) DOK2 1 1 c] f ( x) d] ( h h)( x) x1 e] sketch : CC2 Review 1 * find h ( ) x first
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31 Given f x x and g x x 2 ( ) ( 1) 3 ( ) 3 2 1) ( f g)( 3) 2) ( g f )( 6) 3) ( f g)( x) 4) ( g f )( x) 1 1 5) f ( x) 6) g ( x) 1 7) f(4) and f (12) 1 8) g(5) and g (13) 1 1 9) sketch f ( x) 10) sketch g ( x) 1 What did you notice about 7) and 8)? 1 1 )( )( 1 f f x) 12) (g g )( x) What did you notice about 11) and 12)? 13) Challenge : Sketch ( f g)( x) ) Challenge : if hx ( ) 2 h ( x) 2 ( x 3) CC2 Study Guide Practice
32 For problem 14) you need the parent graph:
33 Practice: Compound interest
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37 Exponential Graphs with b other than 2
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41 Any exponential equation : x y a b where b 1and a is positive where 0 b 1and a is positive i s called exponential growth i s called exponential decay
42 y * 1 x * 1
43 y 2 y 1 -
44 x 2 x 3
45 Parent: y 1 3 x1 2 Parent:
46 Credit to:
47 Find the parent graph, horizontal asymptote, end-behavior, then sketch: Goal Problems Create an exponential equation for each chart: G(x)= DOK 3: Not base 2!!!
48 1] What s the parent? 2] Vertical reflection, stretch, translation 3] Horizontal translation
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50 *3 *3 *3 *3 Method 1: Find the Rate of change first so y a(3) x x Then find a by substitution y (3)
51 Method 2: Pick two pts, (1, 3) and (2, 9) Use the formula y 3 9 ab ab ab Solving equation by substitution a, 9 b, b b b, 3 b 3 Now you found the base : 3 a(3), 1 So the answer : y x 3 a 3 1( 3) x :
52 When you plot the points and sketch, you will notice that there is a horizontal asymptote of -4, this means the y s for the graph before vertical translation down: Method 2: Pick two pts, (1,12) and (2, 48) y ab x 12 ab 48 ab 1 2 solving equation by substitution a, 48 b, b b b, 4 b 12 Now you found the base : a(4), 3 4 x Answer y 3(4) a 4 :
53 Practice: Exponential Graphs Find the parent graph, horizontal asymptote, endbehavior, then sketch: Create an exponential equation for the charts: Create an exponential equation for each graph:
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55 Happy Tuesday Will pass back your quick check tomorrow, sorry No quick check this week It will be next Wednesday Progress Report 3 due next Friday
56 Concept Category 2 Goal Problems for the next quick check: Domain : Range : End behavior : Exponential equation ( dok3)? Given f ( x) 3x 4 find : f g 1 1 g x 2 ( ) ( x 3) 2 ( x) ( x) ( f g)( x) ( g f )( x) Sketch g x and g 1 ( ) ( x)
57 Sketch : x2 1 x2 w( x) 3 5 s( x) Create an exponential equation (dok3)
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63 Happy Wednesday 11/8/17 CC2 Quick Check Error Analysis Go ahead and copy the solution for CC1 Retake, even if you didn t take it Continue working through all of the DOK2 problems from yesterday Before the next Wednesday: YOU NEED to show improvements on CC2, the grade will be computed into Progress Report 3
64 Algebra 1 Review:
65 Nov 13 th Notes Rational Exponents In other words, exponents that are fractions.
66 Definition of b 1 n For any real number b and any integer n > 1, b > 0 1 b n n b
67 Examples:
68 Examples:
69 Definition of Rational Exponents For any nonzero number b and any integers m and n with n > 1, b > 0 m n n m b b b n m
70 There are 3 different ways to write a rational exponent /3 (27 ) 4 (3) 81
71 Examples:
72 Do these:
73 Review:
74 Caution: negative exponent vs. inverse function Base is a constant Base is a variable Base is a function!!! ex) ex) 2 5 x 4 f 1 ( x) x 4 1 x 4 inverse function 1 not f ( x ) switch x and y, solve the y
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77 11/14th Solving exponential equations: P a( b) x Solve for Initial Value: Solve for Base (rate): Solve for Exponent: Ex] 50 a(2) 50 a a a actual equation y (2) 5 x : Ex] (B) 80 9 (B) (B) ( B) B actual equation y 3.5(1.42) 9 B x 9 : 9 Ex] 50 10(2) (2) (2) x x How do you solve for theexponn e t? x
78 Introduction to Logarithmic Functions
79 Logarithmic Functions Review: Inverse of a function Inverse functions is the set of ordered pair obtained by switching the x and y values. f(x) f -1 (x)
80 Logarithmic Functions Review: Inverse Function GRAPHS OF EXPONENTIALS AND ITS INVERSE Inverse functions can be created graphically by a reflection over the line y = x ; or switch the x and y f(x) f -1 (x)
81 Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE A logarithmic function is the inverse of an exponential function Exponential functions have the following characteristics: (0,1) Domain: {x= all real} Range: {y > 0} Horizontal y = 0 Increasing left to right Review: Inverse Function 2 x
82 Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE Let us graph the exponential function y = 2 x Table of values:
83 Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE Let us find the inverse the exponential function y = 2 x Table of values:
84 Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE When we add the function f(x) = 2 x to this graph, it is evident that the inverse is a reflection on the y = x axis f(x) f -1 (x) f(x) f -1 (x)
85 Logarithmic Functions Next, you will find the inverse of an exponential algebraically Write the process in your notes FINDING THE INVERSE OF AN EXPONENTIAL y = a x Interchange x y x = a y base x = a y exponent We write these functions as: x = a y y = log a x y = log a x exponent base
86 Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL y x = a x y log Logarithmic Inverse Exponential of the Form Exponential Function Function
87 Another way to look at the Log formulas: Exponential Form : Logarithmic Form : Result Base Exponent Log Base Result Exponent Calculator : When evaluating formulas: Pay attention to the positions, not the letters or other symbols used to represent the variables Log Result Log Base Exponent
88 Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 3 3 = 27 b) 4 5 = 256 c) 2 7 = 128 d) (1/3) x =27 ANSWERS
89 Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 3 3 = 27 b) 4 5 = 256 c) 2 7 = 128 d) (1/3) x =27 Log 3 27 =3 Log =5 Log =7 log 1/3 27 = x
90 Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log 2 64=6 b) log 25 5=1/2 c) log 8 1=0 d) log 1/3 9=-2 ANSWERS
91 Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log 2 64=6 2 6 = 64 b) log 25 5=1/2 c) log 8 1=0 d) log 1/3 9= /2 = = 1 (1/3) -2 = 9
92 Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log 1/3 27 = x b) log 5 x = 3 c) log x (1/9) = 2 d) log 3 x = 0 ANSWERS
93 Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log 1/3 27 = x b) log 5 x = 3 (1/3) x = 27 (1/3) x = (1/3) -3 x = = x x = 125 c) log x (1/9) = 2 d) log 3 x = 0 x 2 = (1/9) x = 1/3 3 0 = x x = 1
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102 11/20 Happy Monday Sketch : Identify the parents and the a y Horizontal asymptotes x3 ] b y x2 ] x3 x2 1 c] y d] y ) The initial value of your car is $20,000. After 3 years, the value is $ What is the rate of decrease? Find the value of the car at this rate after 7 years. 2) Megan has $28,000 to invest for 5 years at an interest rate of 5.65%. How much money will she have at the end of 5 years if: a] the interest compounds monthly? b] semi-annually? c] quarterly? 3) The initial value of your investment is $55,000. After 18 years, the value is $195,000. What is the rate of increase?
103 Commonly used exponential parent graphs
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107 Nov21 Exponential and Log Equations 1] Log x 4 x? * hint :change to exponential form first 5 2] Log 4 y y? ]12 2( R) R? 4] 33 2(1.5) x? 9 5] 25 3(R) 3 R? 6] x? 7] Log 4 Log 16 Log ] Log (4 16 8) x x
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109 HAPPY MONDAY 11/27 th : Review on Log vs. Exp. Example : Write in log arithmic form. Solution: log Read as: "the log base 2 of is equal to -3". 8
110 2. Write in logarithmic form. Solution: log 5 1 0
111 3. Write in log arithmic form. Solution: log
112 Try these problems Solve these without using your calculator: a) 3 b) 2 c) 2 x x Use your calculator to solve: x a) b) Log 729 c) 25
113 Try these problems Solve these without using your calculator: 3 1 x 1 a) 3 b) 2 c) Log x 4 x 6 Use your calculator to solve: x a) b) Log 729 c) log , x Log x log x log log
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115 Do these problems on your notebook Log Log Log Log Log Log Log 8 Log DOK3: There are patterns to the numbers. Can you find them?
116 Do these problems on your notebook Log Log Log Log Log Log Log 8 Log DOK3: There are patterns to the numbers. Can you find them?
117 Based on the work Log 16 Log 32 Log ( 16 32) Log Log 64 Log 2 Log ( 64 2) Log Log 32 Log 8 Log ( 32 Log Log 4 Log 4 Log ( ) 2 8 )
118 Properties of Logarithms EXPANDED = CONDENSED 1. log a M log a N = log a MN 2. log a M log a N = log a M N 3. r log a M = log ( M ) a r
119 Be Careful! log (a+b) NOT the same as log a + log b log (a-b) NOT the same as log a log b log (a * b) NOT same as (log a)(log b) log (a/b) NOT same as (log a)/(log b) log (1/a) NOT same as 1/(log a) It s all about ( )
120 Use the new formulas Do these problems: Simplify : a] Log 144 Log 16? 3 3 b] Log 12.8 Log 10? 2 2 c] 2Log 10 2Log 5? Solve : 2 2 * HINT : compress to single log function first d] Log 10 Log W e] Log x Log
121 Use the new formulas Do these problems: Simplify : a] Log 144 Log 16? 3 3 b] Log 12.8 Log 10? 2 2 c] 2Log 10 2Log 5? Solve : 2 2 * HINT : compress to d] Log 10 Log W 5 w 3. 2 e] Log x Log 4 3 x single log function first 32
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124 Extra Practice: Solve : 1] Log 5 Log ( x 3) ] Log ( x 5) 2Log ] 2Log x Log
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