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: DOK 3: Not base 2!!!

48 1] What s the parent? 2] Vertical reflection, stretch, translation 3] Horizontal translation

49 *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)

50 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 :

51 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 :

52 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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