Concept Category 2 Exponential and Log Functions
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.
Graphing & Writing an Exponential Growth Model A population of 50 rabbits escaped into a wildlife region. The population triples each year for 8 years.
(Rate)
Writing & Graphing an Exponential Growth Model SOLUTION After 8 years, the population is P = C (3) t Exponential growth model = 50(3) 8 = 50 38 Substitute C, r, and t. Simplify. = 328,050 There will be about 328,050 rabbits after 8 years.
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(1 + 0.07) 15 = 9000(1.07) 15 24,831.28 The value of the painting in 15 years is $24,831.28.
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 2017. b = the original 100% + growth rate y = a(100% + 8%) t = 1200(1 + 0.08) 11 = 1200(1.08) 11 $ 2797.97
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 $ 14557.28
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.?
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).
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 = $1379.85
b) How about $1200 invested at a rate of 2% compounded monthly for 7 years.? b = original 100% + 2% divided by 12 n = 12 0.02 12 t A P 1 12 1200(1 0.00167) 84 1380.55 dollars
Exponential Growth Equations Word Problems (applications)
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 2020. If the profit increased by 12% each year, what would your expected profit be in the year 2032?
Answer Key
Given Parent Graph f ( x) 2 Sketch, find domain, range, end behavior : gx ( ) 1 2 3 w( x) 1 2 x3 x2 x x3 ux ( ) 2 3 s( x) 2 2 x
And then sketch,and find domain, range, end behavior : f g 1 1 1 ( x) ( x) w ( x) The inverse of the Exponential Functions are called Logarithmic Functions u 1 1 ( x) s ( x)
Do you remember all the steps of transformation? Can you find the Asymptotes?
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.
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 : 2 4 3 CC2 Review 1 * find h ( ) x first
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) 5 1 14) Challenge : if hx ( ) 2 h ( x) 2 ( x 3) CC2 Study Guide Practice
For problem 14) you need the parent graph:
Practice: Compound interest
Exponential Graphs with b other than 2
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
y * 1 x * 1
y 2 y 1 -
x 2 x 3
Parent: y 1 3 x1 2 Parent:
Credit to: http://maths.nayland.school.nz/year_12/as_2.2_graphs/8_exponential.htm
Find the parent graph, horizontal asymptote, end-behavior, then sketch: Goal Problems Create an exponential equation for each chart: DOK 3: Not base 2!!!
1] What s the parent? 2] Vertical reflection, stretch, translation 3] Horizontal translation
*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)
Method 2: Pick two pts, (1, 3) and (2, 9) Use the formula y 3 9 ab ab ab 1 1 2 Solving equation by substitution 3 3 2 a, 9 b, b b 9 9 3 b, 3 b 3 Now you found the base : 3 a(3), 1 So the answer : y x 3 a 3 1( 3) x :
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 12 12 2 a, 48 b, b b 48 48 12 b, 4 b 12 Now you found the base : 1 12 12 a(4), 3 4 x Answer y 3(4) a 4 :
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: