Algebra II. Chapter 8 Notes. Exponential and Logarithmic Functions. Name
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1 Algebra II Chapter 8 Notes Eponential and Logarithmic Functions Name
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3 Algebra II 8.1 Eponential Growth Toda I am graphing eponential growth functions. I am successful toda when I can graph eponential growth functions. It is important for me to know/do this because eponential growth functions are used in solving real-life problems. h An eponential function has the general equation of f ab k, where the base, b, is a positive number other than 1. h translates the graph k translates the graph 4 moves moves 4 7 moves moves a 0 (positive), the graph opens a 0 (negative), the graph opens 1. Graph f 2 2. Graph 1 f 2 -intercept asmptote -intercept asmptote as, f as, f as, f as, f
4 . Graph f Graph f 2 4 -intercept asmptote -intercept asmptote as, f as, f as, f as, f 5. Without graphing, describe the difference between the graph of f 2 6 and g 2 7. GROWTH FACTOR When a real-life quantit increases b a fied percentage each ear (or other time period), the amount of the quantit after t ears can be modeled b the equation: t a 1 r a is the initial amount r is the percent increase (written as a decimal) 1 r is the 6. In 1990, the cost of tuition at Faber College was $400. During the net 8 ears, the tuition rose 4% each ear. a. Write a model that gives the tuition (in dollars) t ears after b. Graph the model
5 COMPOUND INTEREST Consider an initial principal P deposited in an account that pas interest at an annual rate r (written as a decimal), compounded n times per ear. The amount A in the account after t ears is modeled b: r AP1 n nt 7. You deposit $1500 in an account that pas 6% annual interest. Find the balance after ears if the interest is compounded: a. annuall b. semi-annuall c. quarterl Homework: 8.1 Worksheet
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7 Algebra II 8.2 Eponential Deca Toda I am graphing eponential deca functions. I am successful toda when I can graph eponential deca functions. It is important for me to know/do this because eponential deca functions are used in solving real-life problems. Eponential deca function have the form of f ab, where 0 a and 0b 1. As increases, decreases. 1. Graph f Graph f intercept asmptote -intercept asmptote as, f as, f as, f as, f 2 2. Graph f 4 5 -intercept asmptote as, f as, f 4. Eplain how the graph of # can be obtained from the graph of #2.
8 5. f 6. f f 2 8. f Growth Growth Growth Growth Deca Deca Deca Deca DECAY FACTOR When a real-life quantit decreases b a fied percentage each ear (or other time period), the amount of the quantit after t ears can be modeled b the equation: t a 1 r a is the initial amount r is the percent decrease (written as a decimal) 1 r is the The deca factor is actuall the percent that remains. 9. There are 40,000 homes in Turdville. Each ear, 10% of the homes are epected to disconnect from septic tanks and connect to the sewer sstem. Write an eponential deca model for the number of home that use septic sstems. 10. A new car costs $2,000. The value decreases b 15% each ear. Write an eponential deca model for the car s value. Use the model to estimate the value after ears. Graph the model. Use the graph to predict when the value will be half of the original value. model value after ears when half of original value Homework: 8.2 Worksheet
9 Algebra II 8. The Number e Toda I am using the number e as the base of eponential functions. I am successful toda when I can use e in real-life situations. It is important for me to know/do this because e is used in solving real-life problems. You have $1 and ou are getting 100% interest per ear (fat chance!). How much will ou have at the end of one ear if the interest is compounded: The natural base e is irrational. It is defined as: 1 As n approaches, 1 n n approaches e **e is used when something grows continuousl** The formula for continuousl compounded interest is: rt Pe, where P is the principal, r is the interest rate and t is the time. 1. In 1982 the population of Bora Bora was 42,000. Population grows continuousl. If the estimated annual growth rate is 2.8%, what will the population be in 2015? 2. You deposit $1500 in an account that pas 6.5% annual interest. What is the balance after 4 ears if it is compounded monthl? continuousl?. 24e 5 6e e e
10 6. e 7. e 0.5 -intercept asmptote -intercept asmptote as, f as, f as, f as, f 8. e intercept asmptote as, f as, f f e Growth Deca 10. f 5e 11. f 4e 7 Growth Deca 12. Growth Deca 1 f e 5 Growth Deca Homework: 8. Worksheet
11 Algebra II 8.4 Logarithmic Functions Toda I am evaluating and graphing logarithmic functions. I am successful toda when I can evaluate and graph logarithmic functions. It is important for me to know/do this because logarithms are used in solving real-life problems A logarithm helps ou find a missing power. The logarithm of with base b is denoted b log b and is read as log base b of. log b is the same as b Super Important let b be a positive real number such that 1 ***log of 1 is alwas zero alwas! b log 1 0 b 6. log log log log log2 11. log log The common logarithm is the logarithm with base 10. It is denoted b log 10 or just log. The natural logarithm is the logarithm with base e. It can be denoted b log e but is tpicall denoted b ln. ***Scientific calculators have kes for evaluating log 10 and ln.*** 1. log7 14. ln 0.25
12 Logarithmic functions are inverses of eponential functions. log b log b b and b Therefore, logarithmic functions and eponential functions undo each other. ***Super Important ln and e undo each other so ln e 1 or 2 ln e log log log ln e log 20. ln log 22. log intercept asmptote -intercept asmptote Homework: 8.4 Worksheet
13 Algebra II 8.5 Properties of Logarithms Toda I am using properties of logarithms to simpl epressions and solve real-life problems. I am successful toda when I can use properties of logarithms. It is important for me to know/do this because logarithms are used in solving real-life problems Properties of Logarithms Product Propert: log uv log u log v (or log u log v log uv ) b b b b b b u u Quotient Propert: logb logb u logb v (or logb ulogb v logb ) v v Power Propert: log b n u nlog u (or nlog u log u b b n b ) 4. log 4 5. log 2 z 6. log log log log log 410 log log8 log85 log log7 5log
14 1. 7 Change of Base Formula log c u log log b More importantl b u c log c u log u log c or log c ln u u ln c ***Change of Base Formula is not necessaril needed if ou have a TI-84 Plus graphing calculator with the updated operating sstem. Those calculators have the abilit to do logs with an base. The TI-8 must use change of base.*** log4 58 Using a TI-84 Plus To get to logbase(, hit ALPHA then F2 Using a TI-8 2 log log5 14. log Homework: page even
15 Algebra II 8.6 Solving Eponential and Logarithmic Equations Toda I am solving eponential and logarithmic equations. I am successful toda when I can solve eponential and logarithmic equations. It is important for me to know/do this because eponential and logarithmic functions are used in solving real-life problems log 51 log 7 *6. log log log e 9. The population of Toic Town is 4,000 and is decreasing b 5% per month. After how man months will there onl b 5,000 people left? 10. Bacteria in a dish grows at a continuous rate of 14% per hour. If there are 10,000 cells now, when will there be 100,000 cells? Homework: 8.6 Worksheet
ab is shifted horizontally by h units. ab is shifted vertically by k units.
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