Math 11A Graphing Exponents and Logs CLASSWORK Day 1 Logarithms Applications

Transcription

1 Log Apps Packet Revised: 3/26/2012 Math 11A Graphing Eponents and Logs CLASSWORK Day 1 Logarithms Applications Eponential Function: Eponential Growth: Asymptote: Eponential Decay: Parent function for Eponential Growth Parent function for Eponential Decay Equation:, where Domain: Range: Equation:, where Domain: Range: Asymptote: Quadrants: Asymptote: Quadrants: Increasing or Decreasing Increasing or Decreasing Eample 1a: Graph y 4 3 Eample 1b: Graph y 4 State the domain, range, and asymptote. State the domain range, and asymptote. 1

2 Log Apps Packet Revised: 3/26/2012 Determine whether the function represents eponential growth or decay. 2. f g h (4) 5. f g 7. h 3 2 Graph the following functions and state the domain, range, and asymptote. 8. y y y Logs Maze - Amaze-ing 2

3 Log Apps Packet Revised: 3/26/2012 Math 11A Graphing Eponents and Logs HOMEWORK D 1 Logarithms Applications Match the function with its graph f( ) f( ) f 1 ( ) 4 1 Graph the following and identify the domain, range, and asymptote y y 3 1 3

4 Log Apps Packet Revised: 3/26/2012 Math 11A Graphing Eponents and Logs CLASSWORK Day 2 Logarithms Applications Warm-up: Find the inverse of the following functions: 1) y = 2-3 2) y = 2-2 New: Find the inverse of the function: 3) y log 2 4) y log 3 2 5) y log 1 6) ylog Switch and y write in B e = N form Solve for y 7) f( ) 3 8) 1 f( ) f( ) f ( ) log 4 4

5 Log Apps Packet Revised: 3/26/2012 Parent Graphs for Logarithmic Functions Equation:, where Domain: Range: Asymptote: Quadrants: Equation:, where Domain: Range: Asymptote: Quadrants: log 11. Graphing logarithmic functions **Use COB logb log b a. y log 2 b. y log 1 3 log 12. Graphing logarithmic functions **Use COB logb log b a. y log3 b. y log 1 2 5

6 Log Apps Packet Revised: 3/26/2012 Math 11A Graphing Eponents and Logs HOMEWORK Day 2 Logarithms Applications Tell whether the function represents eponential growth or eponential decay f( ) f( ) 4 3. f( ) 2 Match the function with its graph. 4. f f f A B C Graph the function. State the domain and range 7. f ( ) log3 8. y log 1 4 d: r: d: r: Find the inverse of the following equations: 9. y log5 10. y log 1 `11. y 2 7 6

7 Log Apps Packet Revised: 3/26/2012 Log Applications Classwork Day 3 Graphing Logs and Eponential Equations 1. Which is the equation of the graph shown? 1) y = 2 2) y = 10 3) y = log 2 4) y = log Which is an equation of the graph shown? 1) y = log 2 2) y = -log 2 3) y = 2 4) y = 2-3. The graph of y = a, where a > 0, contains the point 1) (1, 0) 2) (-1, 0) 3) (0,0) 4) (0, 1) 4. The graph of y = 1 2 lies entirely in Quadrants 1) I and II 2) I and III 3) II and III 4) I and IV 5. Which statement accurately describes the graph of y = 10. 1) It is an increasing function and lies entirely in Quadrants I and II. 2) It is an increasing function and lies entirely in Quadrants I and IV. 3) It is a decreasing function and lies entirely in Quadrants I and II. 4) It is a decreasing function and lies entirely in Quadrants I and IV. 6. The graph of the equation y = 2 intersects 1) the -ais only 2) the y-ais only 3) the -ais and y-ais 4) neither the -ais nor the y-ais 7

8 Log Apps Packet Revised: 3/26/ Given the function y = 2, epress the value of y if = Which is an equation of the inverse of the function y = 2? 1) y = log 2 2) y = 2 3) = y 2 4) = log 2 y 9. What is the -intercept of the graph of the equation y = log 2? 1) 1 2) 2 3) 0 4) What statement is true for the graph of y = log 10? 1) It is an increasing function and lies entirely in Quadrants I and II. 2) It is an increasing function and lies entirely in Quadrants I and IV. 3) It is a decreasing function and lies entirely in Quadrants I and II. 4) It is a decreasing function and lies entirely in Quadrants I and IV. 11. If the graph of the equation of y = 3 is reflected in the -ais, the equation of the reflection is 1.) y = 3-2) y = -(3 ) 3) y = log 3 4) y = If the graph of y = 1 and y = log are sketched on the same aes, then the number of points of in their intersection is 1) 1 2) 2 3) 3 4) 0 13 a) Sketch and label the graph y 3 b) Sketch the inverse of this graph c) Write the equation of the graph in b 8

9 Log Apps Packet Revised: 3/26/ A Logs Applications HOMEWORK 3 Graphing Logs and Eponential Equations Write the inverse of each equation. 1. f( ) 2 2. f ( ) log3 3. y 6 4. y log7 5. f( ) f ( ) log9 In 7-14 Select the numeral preceding the epression that best completes the sentence or answers the question. 7. The graph of y log8 lies entirely in quadrants: 1) I and II 2) II and III 3) I and III 4) I and IV 8. The function that is the inverse of y = 5 is: 1) y 5 2) y 5 3) y log5 4) log5 y 9. At what point does the graph of y log5 intersect the -ais? 1) (1, 0) 2) (0, 1) 3) (5, 0) 4) There is no point 10. A number that is not in the domain of the function y log10 1 1) 1 2) 0 3) 2 4) If y log10 and > 1, then y is: 1) positive 2) zero 3) negative 4) not a real number 12. If y log10 and = 1, then y is: 1) positive 2) zero 3) negative 4) not a real number 13. If y log10 and 0 < < 1, then y is: 1) positive 2) zero 3) negative 4) not a real number 14. If y log10 and 0, then y is: 1) positive 2) zero 3) negative 4) not a real numbera 9

10 Math 11A Graphing Growth and Decay Classwork Day 4 Logarithms Applications where Y A Eponential Growth and Decay Word Problems Y = A(1 ± r) t (1 + r) (1 r) r t Write an eponential growth or decay model that describes the situation. State the initial amount, the growth or decay factor, and % of growth or decay. 1. Population Growth: In Colorado in 1980 the population was 2.8 million. The population has been increasing since then at a rate of 1.5% per year. Initial amount = r = Growth or decay factor: Equation: 2. Economy: Amount of fuel used annually by each car in the US has decreased by 2% each year since 1980, when it was about 591 gallons. Initial amount = r = Growth or decay factor: Equation: 3. Investments: originally invested $48,000 and it grows at an annual rate of 5% compounded yearly. Initial amount = r = Growth or decay factor: Equation: 4. Sports: Average salary for professional baseball players in the US is increasing annually at a rate of 22% since 1984 when the starting salary was $283,000. Represent salary in the model as thousands of $. Initial amount = r = Growth or decay factor: Equation: 1Log Apps Packet Revised: 3/26/2012 0

11 Multi-step Problems 5. In 2003 there were 38 buffalo in a state park and that number grew by about 7% per year. a. Write an eponential model giving the number n of buffalo after t years. About how many buffalo were in the park after 7 years? b. Graph the model. Use the graph to estimate the year in which the buffalo population reached A new television costs $1200. The value of the television decreases by 21% each year. a. Write an eponential decay model giving the television's value y (in dollars) after t years. Estimate the value after 2 years. b. Graph the model. Use the graph to estimate when the value of the television will be $ You buy a new car for $22,500. The value of the car decreases by 25% each year. a. Write an eponential decay model giving the car's value V (in dollars) after t years. What is the value of the car after four years? b. In approimately how many years is the car worth $10,000? 1Log Apps Packet Revised: 3/26/2012 1

12 Math 11A Graphing Growth and Decay Logarithms Applications Tell whether the following are eponential growth or decay 1. 3 f( ) f( ) 5 3. g ( ) 4 HOMEWORK D4 Write the inverse of the following equations y log 4 5. y 3 6. y 2 7. Population: From 1990 to 2000, the population of California in millions of people can be modeled by P = 29(1.0128) t, where t is the number of years since a) What was the population in (in millions) in 1990? b) What is the growth factor and annual percent increase? c) Estimate the population (in millions) in Depreciation: You buy a new car for $22,500. The value of the car decreases by 25% each year. a) Write an eponential decay model giving the car's value V (in dollars) after t years. b) What is the value of the car after three years? c) In approimately how many years is the car worth $5300? 1Log Apps Packet Revised: 3/26/2012 2

13 Math 11A Growth and Decay Word Problems Classwork D5 Logarithms Applications Write an eponential growth or decay model that describes the situation. State the initial amount, the growth or decay factor, and % of growth or decay; then answer the questions 1. Population Growth: In Colorado in 1980 the population was 2.8 million. The population has been increasing since then at a rate of 1.5% per year. a) Find the population in Colorado in b) Find the year when the population is 3.5 million. Initial amount = r = Growth or decay factor: Equation: 2. Economy: Amount of fuel used annually by each car in the US has decreased by 2% each year since 1980, when it was about 591 gallons. a) Find the amount of fuel in the year b) Find the year during which the amount of fuel was 200 gallons. Initial amount = r = Growth or decay factor: Equation: 3. Investments: $48,000 is invested in 1997 and it grows at an annual rate of 5% compounded yearly. a) Find the year in which the investment doubles b) Find the amount in the year 2015 Initial amount = r = Growth or decay factor: Equation: 1Log Apps Packet Revised: 3/26/2012 3

14 4. Sports: Average salary for professional baseball players in the US is increasing annually at a rate of 22% since 1984 when the starting salary was $283,000. Represent salary in the model as thousands of $. a) Find the year when pro baseball players started earning one million dollars. b) Find the starting salary in 2010 Initial amount = r = Growth or decay factor: Equation: 5. In 2003 there were 38 buffalo in a state park and that number grew by about 7% per year. Write an eponential model giving the number n of buffalo after t years. a) About how many buffalo were in the park after 7 years? b) In what year did the population of buffalo double? 6. A new television in 2006 costs $1200. The value of the television decreases by 21% each year. a. Write an eponential decay model giving the television's value y (in dollars) after t years. Estimate the value after 2 years. b. Find the year when the TV's value is one fourth its original value. 1Log Apps Packet Revised: 3/26/2012 4

15 Math 11A Growth and Decay Logarithms Applications HOMEWORK D5 1. Suppose $320 is deposited in a bank account, earning 7% interest, compounded annually. a. Write an equation modeling the growth of the $320 over a period of t years. b. Find the amount, to the nearest cent, in the account on 6/24/2007 if the initial deposit was made on 6/24/2000. c. Find the first year in which the amount saved will be at least $ An investment of $5000 was made in the stock market in January Unfortunately, it depreciated 14% per year. a. How much money would you epect to have in January 2008? b. Find the first year in which the amount invested depreciates to $ The number of Canadian geese roaming the sports fields in one school district grows every year according to the function G(t) = 24(1.2314) t where t represents the number of years since The PTA has suggested hiring dogs to chase the geese, but the dog squads will not work with geese populations fewer than 70. In what year can the county hire the geese chasers? 1Log Apps Packet Revised: 3/26/2012 5

16 Math 11A Growth and Decay Logarithms Applications Classwork D6 1. A dastardly criminal planted 37 ounces of radioactive krypton-85 in a safety deposit bo near the Daily Planet on January 1, The function showing the strength of the remaining krypton-85 is K(t) = 37(0.9376) t where t represents the years since If any amount over 8 ounces of krypton will immobilize Superman, a. What is the first year in which there will be fewer than 8 ounces of krypton in the bank vault? b. Will it be safe for Superman to enter the bank vault to prevent a planned jewel heist on New Years Eve, 2004? c. How much krypton-85, to the nearest tenth of an ounce, remains on January 1, 2004? 2. Depreciation (the decline in cash value) on a car can be determined by the formula V C( 1 r) t, where V is the value of the car after t years, C is the original cost, and r is the rate of depreciation. If a car s cost, when new, is $15,000, the rate of depreciation is 30%, and the value of the car now is $3,000, how old is the car to the nearest tenth of a year? 1Log Apps Packet Revised: 3/26/2012 6

17 3. An archaeologist can determine the approimate age of certain ancient specimens by measuring the amount of carbon-14, a radioactive substance, contained in the specimen. The formula used to t determine the age of a specimen is 5760 A A0 2, where A is the amount of carbon-14 that a specimen contains, A 0 is the original amount of carbon-14, t is time, in years, and 5760 is the halflife of carbon-14. A specimen that originally contained 120 milligrams of carbon-14 now contains 100 milligrams of this substance. What is the age of the specimen, to the nearest hundred years? t 4. The equation for radioactive decay is p ( 05. ) H, where p is the part of a substance with half-life H remaining radioactive after a period of time, t. A given substance has a half-life of 6,000 years. After t years, one-fifth of the original sample remains radioactive. Find t, to the nearest thousand years. 5. Sean invests $10,000 at an annual rate of 5% compounded continuously, according to the formula A Pe rt, where A is the amount, P is the principal, e = 2.718, r is the rate of interest, and t is time, in years. a) Determine, to the nearest dollar, the amount of money he will have after 2 years. b) Determine how many years, to the nearest year, it will take for his initial investment to double. 1Log Apps Packet Revised: 3/26/2012 7

18 Math 11A Growth and Decay Log Application Homework D6 1. The amount A, in milligrams, of a 10-milligram dose of a drug remaining in the body after t hours is t given by the formula A 10( 08. ). Find, to the nearest tenth of an hour, how long it takes for half of the drug dose to be left in the body The growth of bacteria in a dish is modeled by the function f ( t) 2. For which value of t is f ( t) 32? (1) 8 (2) 2 (3) 15 (4) 16 t 3. Growth of a certain strain of bacteria is modeled by the equation 2, 500 4( 2. 7) t, where: G = final number of bacteria A = initial number of bacteria t = time (in hours) In approimately how many hours will 4 bacteria first increase to 2,500 bacteria? Round your answer to the nearest hour. 1Log Apps Packet Revised: 3/26/2012 8

19 Math 11A Review Log Application Classwork D7 1. The current population of Little Pond, New York, is 20,000. The population is decreasing, as t represented by the formula P A(1.3), where P = final population, t = time, in years, and A = initial population. a) What will the population be 3 years from now? Round answer to the nearest hundred people. b) To the nearest tenth of a year, how many years will it take for the population to reach half the present population? 2. Which equation models the data in the accompanying table? Time in hours, Population, y (1) y 2 5 (2) y 2 (3) y 2 (4) y 5(2 ) 3. What is the domain of f ( ) 2? (1) all integers (2) all real numbers (3) 0 (4) 0 4. The strength of a medication over time is represented by the equation y 200( 15. ), where represents the number of hours since the medication was taken and y represents the number of micrograms per millimeter left in the blood. Which graph best represents this relationship? 1Log Apps Packet Revised: 3/26/2012 9

20 5. Using a 4 bo a) sketch the graph of f ( ) log3 1 f b) on the same grid, graph 1 c) write the equation of f 6. The graphs of the equations y 2 and y 2 a intersect in Quadrant I for which values of a? (1) 0 a 1 (2) a 1 (3) a 1 (4) a 1 7. Are the following eponential growth or eponential decay a) 2 f( ) 5 b) f( ) 2 c) 3 f( ) 2 8. Write the inverse of the following equations. 1 a) y 5 b) y 2 c) y log 4 2Log Apps Packet Revised: 3/26/2012 0

21 Math 11A Growth and Decay Logarithms Applications HOMEWORK D7 0.1m 1. After an oven is turned on, its temperature, T, is represented by the equation T 100(3.2), where m represents the number of minutes after the oven is turned on and T represents the temperature of the oven, in degrees Fahrenheit. How many minutes does it take for the oven's temperature to reach 350 F? Round your answer to the nearest minute. 2a. Sketch the graph of f() = 2 b. On the same set of aes, sketch the graph of f -1 () c. Write the equation of f -1 () 3. Are the following eponential growth or eponential decay a) f( ) 6 b) 1 f( ) 6 c) 7 f( ) 2 4. Write the inverse of the following equations. y log b) y log7 c) y 6 a) 1 4 2Log Apps Packet Revised: 3/26/2012 1