8 Sections 4. & 4.2 Eponential Growth and Eponential Deca What You Will Learn:. How to graph eponential growth functions. 2. How to graph eponential deca functions. Eponential Growth This is demonstrated b the classic riddle in which a child is offered two choices for an increasing weekl allowance: the first option begins at cent and doubles each week, while the second option begins at $ and increases b $ each week. Although the second option, growing at a constant rate of $/week, pas more in the short run, the first option eventuall grows much larger: Eponential Growth! The equation for option is: = 2 n where n is the number of weeks. The equation for option 2 is = + n where n is the number of weeks Eponential Growth Equation: An eponential function involves the epression b where the base than. b is a positive number other The variable is going to be in the position of the eponent.
86 Let s Graph an Eample = 2 - Question: Will the graph ever pass below of? We sa that there is an asmptote at =. An asmptote is a line that a graph approaches as ou move awa from the origin. ab is an eponential growth function when a is greater than and b is greater than. It is important to know the ordered pairs to get our starting points for our graph. The will alwas be,,(,), and (, b). b
87 In the eponential model, a, b. If we transform f ( ) ab f ( ) ab f ( ) a. So the b eponential growth function reflects over the -ais (multipling all -values b -). QUESTION? Is it still eponential growth? Let s Graph an Another Eample 2 2 -
88 Graphing b Transformation: The generic form of an eponential function is: ab h k Where h is movement along the -ais and k is movement along the -ais. The value of a will give a reflection and/or a vertical stretch or compression. The value of b will give a horizontal reflection when b is negative.(new Transformation!) An Eample of Graphing b Translation Graph 2 2 3 - - - - a. State the domain and range. D: R: b. Give the equation of the horizontal asmptote. HA: - Graph 3 2 4 - - - - a. State the domain and range. D: R: b. Give the equation of the horizontal asmptote. HA: -
89 Application:. State whether the following model eponential growth or eponential deca. 2. Describe the transformation of the parent function. 3. State the domain and range. 4. Identif the -intercept and the asmptote. 3(2). Growth Deca 2. 3. D: R: 4. -int: Asmptote: 2 6( ) 4. Growth Deca 2. 3. D: R: 4. -int: Asmptote: 2( ) 4. Growth Deca 2. 3. D: R: 4. -int: Asmptote: 4 2. Growth Deca 2. 3. D: R: 4. -int: Asmptote: 2 9. Growth Deca 2. 3. D: R: 4. -int: Asmptote: An Eponential Deca Word Problem: When a real=life quantit increases or decreases b a fied percent each ear (or other time period) the amount of of the quantit after t ears can be modeled b the formula: a( r) t ( r) is called the growth factor ( r) is called the deca factor
9 Eample: You bu a new car for $24,. The value of the car decreases b 6% each ear. Initial amount is: Annual % increase or decrease Growth or deca? Growth/deca factor: a. Write an eponential model for the value of the car. b. Use the model to estimate the value after 2 ears. c. When will the car have a value of $2,? In Januar 993, there were about,33, Internet hosts. During the net five ears, the number of hosts increased b about % per ear. Initial amount is: Annual % increase or decrease Growth or deca? Growth/deca factor: a. Write an eponential model for the number of hosts. b. Use the model to estimate the number of hosts in 996. c. When will the number of hosts reach 3 million? Another eample: A motor scooter purchased for $ depreciates at an annual rate of %. a. Write an eponential function and graph the function. b. Use the function to predict when the value will fall below $.
9 Eample: Your parents want to deposit $ in the bank as an investment for ou the da ou are born! The will give ou the mone when ou turn 8. The banks the can use all pa a earl interest rate of 6.%, but compound the interest differentl. Which bank should the choose to invest their mone in? Bank A: Interest Compounded Annuall: Bank D: Interest Compounded Monthl: Bank B: Interest Compounded Semi-Annuall: Bank E: Interest Compounded Dail: Bank C: Interest Compounded Quarterl:
92 Section 4.3 The Number e The number e is known as Euler s number. Leonard Euler (7 s) discovered its importance. An irrational number, smbolized b the letter e, appears as the base in man applied eponential functions. It models a variet of situations in which a quantit grows or decas continuousl: mone, drugs in the bod, probabilities, population studies, atmospheric pressure, optics, and even spreading rumors! The number e is defined as the value that approaches as n gets larger and larger. n,,,,,, As n, e 2.78 n n Eample : Simplifing the Natural Base Epressions n n a) e e b) 3 4 e 2 e 3 4 2 c) (3 e ) d) (2 ) 2 e Eample 2: Use a calculator to Evaluate the Epressions a) 6 e b) e.2 Graphing e: Since 2 < e < 3, the graph of = e is between the graphs of = 2 and = 3
93 The irrational number e, is called the natural base. The function eponential function. f e is called the natural - Graph: f ( ) e - - - a. State the domain and range. D: R: b. Give the equation of the horizontal asmptote. HA: - Graph: f ( ) e - - - - - a. State the domain and range. D: R: b. Give the equation of the horizontal asmptote. HA:
94 Graph e 3 - - - - a. State the domain and range. D: R: b. Give the equation of the horizontal asmptote. HA: - Graph 2 e - - - - - a. State the domain and range. D: R: b. Give the equation of the horizontal asmptote. HA:
9 Continuous Compound Interest Eample: John invests $7 into a bank account that pas at a rate of.6% compounded continuousl. Assuming he does not add an more mone to the account, how much mone will be in his account in ears? How much after 3 ears? ears 3 ears Eponential Growth The function kt A( t) A e, where k can model man kinds of population growths. Where: A = population at time, At () = population after time, t = amount of time, k = eponential growth rate. The growth rate unit must be the same as the time unit. Eponential Deca Deca, or decline, is represented b the function kt A( t) A e, where k Where: A = initial amount of the substance, t = amount of time, At () = amount of the substance left after time, k = eponential deca rate. The growth rate unit must be the same as the time unit. The half-life is the amount of time it takes for half of an amount of substance to deca.
96 Eponential Model for the Half-Life of a substance: Use the graph to determine the half-life and the eponential model for Iodine-3. The inverse function of as ln. e is the natural log and is a log of base e. It can be written as log e or simpl [ANYTIME ln and e ARE NEXT TO EACH OTHER, THEY WILL CANCEL THEMSELVES OUT.] lne = lne = lne Use the properties above to solve the following equations. 2 a. e 2. 2 b. e 3 c. 2 e 7.2 9. d. 2 6 2 4 6 2 3 2 e. 3 f.
97 Continuous Interest: Eample : An initial investment of $ is now valued at $49.8. The interest rate is 8% compounded continuousl. How long has the mone been invested? Eample 2: An initial investment of $2 is now valued at $3.24 after compounding continuousl for 7 ears. Find the interest rate. Eample 3: What interest rate do ou need for a $ investment to double in ears? Eample 4: Carbon-4 is used to determine the age of artifacts in carbon dating. It has a half-life of 73 ears. Write the eponential deca function for a 24 mg sample and then find the amount of carbon-4 remaining after 3 millennia. ( millennium = ears.)
98 Sections 4.4-4.6 Logarithmic Functions The inverse function to an eponential function is the logarithm. If ou are tring to solve an eponential and ou get stuck, change it to a logarithm and vice versa. A Logarithm to the base of a positive number is defined as follows: Eponential form Logarithmic Form b is the same as log b = log b is read as The log, base b, of. The -part of a log is called the argument of the log. The epression log b represents the eponent to which the base b must be raised in order to obtain. The domain of a logarithm can be found b setting the argument greater than zero and solving. Find the domain: a. f ( ) log 3(2 ) b. f 2 ( ) ln( 4) Changing a log to an eponential: just remember that THE BASE IS THE BASE IS THE BASE... Remove the log operator and switch the and s position. Changing an eponential to a log still remember that THE BASE IS THE BASE IS THE BASE. Introduce the log operator and make the base of the eponential the base of the log, then switch the and s position. Eponential Form 2 3 9 Logarithmic Form log 2 2 log 2 2 log8 Special Properties of logs: a. logb b. log b b c. log b p b p d. log b p b p
99 Evaluate a. log 4 64 b. log2 c. log 26 d. 8 4 log3 3 e. log7 7 f. log3 A common log is a log of base. It can be written as log or simpl as log because the base of a log that has no base showing is understood to be base. A natural log is a log of base e. It can be written as log e or simpl as ln. Evaluate using the Change of base formula. Did ou get the same answers as earlier? a. log 4 64 b. log2 c. log 26 d. log7 7 f. log3 8 4 Use log9.732 and log9.9 to approimate the following without a calculator. a. log9 b. log9 c. log9 2 Epand: a. log 6 log 2 b. 7 3
Condense: a. log 6 2log 2 log3 b. 2log8 log8 log8 3 Solving equations (Solutions can be EXTRANEOUS) a. log 4( 3) 2 b. log 6( ) log6 2 c. d. 8 2 e. ln(3 9) 2 f. log 4( 3) log 4(8 7) g. 2 32 4 h. 4 8 3 i. 2 4 6 j. 2 3 3 Eponential word problem (revisited): Recall the car depreciation problem from earlier in the unit: 24,(.6) t. We used the calculator intersect to solve the problem of when the car would be worth $2,. Let s solve it now without a calculator using what we have learned about solving eponential equations.
Word Problems. The slope s of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach b this equation s.9.8log d. Find the slope of the beach if the average diameter of the sand particle is.2 mm. 2. The Richter magnitude M of an earthquake is based on the intensit I of the earthquake and the intensit I of an earthquake that can be barel felt. One formula used is M log I. If the I 6.8 intensit of the Los Angeles earthquake in 994 was times I, what was the magnitude of the earthquake? What magnitude on the Richter scale does an earthquake have if its intensit is times the intensit of a barel felt earthquake? 3. The moment magnitude M of an earthquake that releases energ E (in Ergs) can be modeled b the equation M.29ln E.7. If the earthquake in Prince William Sound in 964 had a moment magnitude of 8.6, how much energ did it release?
2 Section 4.4 Graphing and Modeling Eponential and logarithmic functions are inverses of each other. The graph of = 2 is shown in red. The graph of its inverse is found b reflecting the graph across the line =. LOGARITHMIC FUNCTION f() = log b Domain: (, ) Range: (, ) () = log b, a >, is increasing and continuous on its entire domain, (, ). The -ais is a vertical asmptote as from the right. The graph passes through the points (, ), (,), and (b, ) b Eample: Graph the function log 3 - - - - -
3 You can translate the graph of logarithms using the rule previousl discussed applied to the equation alog ( h) k. b How does a affect the graph? How does h affect the graph? How does k affect the graph? Now graph: log ( ) 3 - a. State the domain and range. D: R: b. Give the equation of the vertical asmptote. VA: - - - - log ( 6) 4 3 - - - a. State the domain and range. D: R: b. Give the equation of the vertical asmptote. VA: - -
4 log ( 3) 2 - - - - a. State the domain and range. D: R: b. Give the equation of the vertical asmptote. VA: - You can also graph log base and log base e (ln) in our graphing calculator. Show how ou could use the Change of Base formula to graph the same functions in our calculator. a. log3 b. log 3( ) c. log 3( 6) 4 d. log 2( 3) a. b. c. d. Now graph letter d in our calculator. Is it the same graph ou drew b hand earlier? Y N
Logistic Growth A logistic growth model is a function of the form f() = +ae r where a, r, and c are positive constants. The number c is called the carring capacit. Logistic functions, like eponential functions grow quickl at the beginning, but because of restrictions that place limits on the size of the underling population, eventuall grow slowl and then level off. c Domain: (, ) Range: (, c ) Eample:
6 Evaluating a Logistic Growth Function Eample: Evaluate the function f( ) for the given value of..3 e a. f b. f 4 c. f Solving a Logistic Growth Equation: Eample: Solve the equations. a. 2 2e 2 4 b. 4 e c. 3 e Using a Logistical Growth Model:
7 Regression Revisited:. A population of single-celled organisms was grown in a Petri dish over a period of 6 hours. The number of organisms at a given time is recorded in the table shown. a. Determine the eponential regression equation model for these data, rounding all values to the nearest ten-thousandth. b. Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 8 th hour. 2. The given data shows the average growth rates of 2 Weeping Higan cherr trees planted in Washington, D.C. At the time of planting, the trees were one ear old and were all 6 feet in height. a. Determine a logarithmic regression model equation to represent this data. b. What was the average height of the trees at one and one-half ears of age? (to the nearest tenth of a foot) c. If the height of a tree is 2 feet, what is the age of the tree to the nearest tenth of a ear?