Unit 7 Exponential Functions. Mrs. Valen+ne Math III

Transcription

1 Unit 7 Exponential Functions Mrs. Valen+ne Math III

2 7.1 Exponential Functions Graphing an Exponen.al Func.on Exponen+al Func+on: a func+on in the form y = ab x, where a 0, b > 0 and b 1 Domain is all real numbers Range of a parent func+on is y > 0 To graph an exponen+al func+on, make a table of points, plot them, and connect in a smooth curve. Example: What is the graph of y = 2 x?

3 7.1 Exponential Functions Iden.fying Exponen.al Growth and Decay Exponen+al Growth Value of x increases as y increases b is growth factor (b > 1) Exponen+al Decay value of x increases as y decreases b is decay factor (0 < b < 1) In both cases, y-intercept is (0,a), the domain is all real numbers, the range is y > 0, and the asymptote is y = 0. Examples: Which are growth and which are decay? y = 12(0.95) x decay y = 0.25(2) x growth You put $1000 into a college savings account for four years. The account pays 5% interest annually. à growth

4 7.1 Exponential Functions Modeling Exponen.al Growth For exponen+al growth and decay, b = 1 + r. If r > 0, it is the rate of increase or growth rate. If r < 0, it is the rate of decay. Exponen+al growth and decay can be modeled by: Example: You put $1000 into a college savings account for four years. The account pays 5% interest annual interest. How much money will be in the account ayer six years? a = 1000 r = 0.05 t = 6 A(6) = 1000 ( ) 6 A(6) = 1000 (1.05) 6 A(6) = $

5 7.1 Exponential Functions Using Exponen.al Growth Suppose you invest $1000 in a savings account that pays 5% annual interest. If you make no addi+onal deposits or withdrawals, how many years will it take for the account to grow to at least $1500? a = 1000 r = 0.05 A(t) ( ) t The account will not contain $1500 un+l the ninth year. AYer nine years, the balance will be $ Suppose you invest $500 in a savings account that pays 3.5% annual interest. When will the account contain at least $650?

6 7.1 Exponential Functions Wri.ng an Exponen.al Func.on Exponen+al func+ons are oyen discrete. To model a discrete solu+on using an exponen+al func+on in the form y = ab x, find the growth or decay factor, b. If you know y values for two consecu+ve x-values, you can find the rate of change, and therefore b. b = 1 + r The equa+on for the graph provided: = b = b = a = 1000 (y-intercept) y = 1000(1.045) x

7 7.1 Exponential Functions Example: The table shows the world popula+on of the Iberian lynx in 2003 and If this trend con+nues and the popula+on is decreasing exponen+ally, how many Iberian lynx will there be in 2014? b = b = 0.8 = = a(0.8) 0 (x = 0 for ini+al value) 150 = a y = 150(0.8) x x = = 11 y = 150(0.8) 11 y 13 Iberian lynx ley in 2014

8 7.2 Properties of Exponential Functions Graphing y = ab x. Recall that the factor a in y = ab x can stretch or compress, and possibly reflect the graph of the parent func+on y = b x. Example: graph y = 2 x and y = 3*2 x. The red graph is the parent func+on while the blue graph is the stretched func+on. Since 3 is posi+ve, the graph is not reflected on the x-axis.

9 7.2 Properties of Exponential Functions Transla.ng the Parent Func.on y = b x. Horizontal y = ab (x-h) shiys the func+on ley or right h spaces Ver+cal y = ab x + k shiys the func+on up or down k spaces Examples: Compare the graph of each func+on to its parent func+on. y = 2 (x 4) y = 20 (½) x + 10

10 7.2 Properties of Exponential Functions Using an Exponen.al Model All transforma+ons combined: y = ab (x h) + k Example: The best temperature to brew coffee is between 195 F and 205 F. Coffee is cool enough to drink at 185 F. The table shows temperature readings from a sample cup of coffee. How long doe it take for a cup of coffee to be cool enough to drink? Use an exponen+al model. Room temp: 68 F Plot the points in the calculator using STAT. In STAT, L3 = L2-68 Use ExpReg L1, L3 to find the exponen+al model. Translate the model ver+cally by 68 y = (0.956) x + 68 It will take about 3.1 min for the coffee to cool to 185 F

11 7.2 Properties of Exponential Functions Rewri.ng an Exponen.al Func.on The func+on P = 80(1.25) d models the popula+on of a city, in thousands, ayer d decades. What exponen+al func+on models the popula+on ayer t years? What is the annual growth rate of the city s popula+on? 10d = t, or d = 0.1t P = 80(1.25) 0.1t P = 80( ) t P = 80(1.023) t A(t) = a(1 + r) t b = 1 + r = 1 + r = r 2.3% is the annual growth rate

12 7.2 Properties of Exponential Functions Evalua.ng e x. Some+mes, a func+on can have an irra+onal base. The graph of has an asymptote of y = e. Recall that e Natural base exponen+al func+ons are exponen+al func+ons with base e. Used to describe con+nuous growth or decay. Otherwise have the same proper+es as other exponen+al func+ons. Examples: Evaluate the following: e 3, e 6, and e e. e e e e

13 7.2 Properties of Exponential Functions Con.nuously Compounded Interest The formula for con+nuously compounded interest: Example: Suppose you won a contest at the start of 5 th grade that deposited $3000 in an account that pays 5% annual interest compounded con+nuously. How much will you have in the account when you enter high school 4 years later? Express the answer to the nearest dollar. A = Pe rt A = 3000e (0.05)(4) A = 3000e (0.2) A $3664

14 7.3 Logarithmic Functions as Inverses Wri.ng Exponen.al Equa.ons in Logarithmic Form Logarithm base b of posi+ve number x: For b > 0, b 1, log b x = y if and only if b y = x Domain: x > 0 This is the inverse of exponen+al func+ons. Example: what is the logarithmic form of each equa+on? 100 = =3 4 x = b y then log b x = y log = 2 x = b y then log b x = y log 3 81 = 4

15 7.3 Logarithmic Functions as Inverses Evalua.ng a Logarithm What is the value of log 8 32? Write a logarithmic equa+on Convert to exponen+al form Re-write using like bases Use Power Property of Exponents Set the exponents equal to each other (bases are same) Solve for x log 8 32 = x 32 = 8 x 2 5 = (2 3 ) x 2 5 = 2 3x 5 = 3x 5/3 = x

16 7.3 Logarithmic Functions as Inverses Using a Logarithmic Scale Common logarithm is log 10 (can be wriuen without base) Some+mes, measures of physical phenomena have such wide range values that the values reported are logarithms of the values. When you use the logarithm of a quan+ty instead of the quan+ty, you are using a logarithmic scale (Ex Richter scale).

17 7.3 Logarithmic Functions as Inverses Example: In December 2004, an earthquake with magnitude 9.3 on the Richter scale hit off the northwest coast of Sumatra. In March 2005, another one hit Sumatra with magnitude 8.7. The formula below compares the intensity levels of earthquakes where I is the intensity level determined by a seismograph, and M is the magnitude on a Richter scale. How many +mes more intense was the December earthquake than the March earthquake?

18 7.3 Logarithmic Functions as Inverses Graphing a Logarithmic Func.on Logarithmic func+on is the inverse of an exponen+al func+on. Recall that the graphs of inverse func+ons are reflec+ons of each other across y=x. Example: What is the graph of y = log 3 x? Describe the domain and range and iden+fy the y-intercept and asymptote. Domain is x > 0. The range is all real numbers. No y-intercept. Asymptote is x = 0.

19 7.3 Logarithmic Functions as Inverses Transla.ng y = log b x Transformed version: y = a log b (x h) + k a stretches ( a >1) or shrinks ( a <1), and reflects (-) over x- axis. h translates horizontally. k translates ver+cally Example: How does the graph of y = log 4 (x 3) + 4 compare to the graph of the parent func+on?

20 7.4 Properties of Logarithms Simplifying Logarithms Proper+es of logarithms are derived from proper+es of exponents. Examples: What is each expression wriuen as a single logarithm? log 4 32 log 4 2 6log 2 x + 5 log 2 y = log = 2 = log 4 16 = log 2 x 6 + log 2 y 5 = log 2 x 6 y 5

21 7.4 Properties of Logarithms Expanding Logarithms These proper+es can be used to expand a single logarithm. Examples: What is each logarithm expanded? = log 4x log y = log 4 + log x log y = log 9 x 4 log = 4log 9 x log = log log 3 (3x 3) 2 = log log 3 (3x 3) = log log log 3 (x 1) = log 8 8 (3a 5 ) ½ = log log 8 (3 ½ a 5/2 ) = log log 8 3 ½ + log 8 a 5/2 = 1+ ½log / 2 log 8 a

22 7.4 Properties of Logarithms Using the Change of Base Formula Since the calculator only uses the common logarithm, you will need to be able to change bases to evaluate some of the logarithms. Examples: What is the value of each expression? log log 5 36 =

23 7.4 Properties of Logarithms Using a Logarithmic Scale The ph of a substance equals log [H + ], where [H + ] is the concentra+on of hydrogen ions. [H + a ] for household ammonia is [H + v ] for vinegar is 6.3 x What is the difference of the ph levels of ammonia and vinegar? ph = log [H + ] log [H + a ] ( log [H+ v ]) = log [H + a ] + log [H+ v ] = log [H + v ] log [H+ a ] = log (6.3x10 3 ) log = log log 10 3 log = log ( 3log 10) ( 11log 10) = log

24 7.5 Exponential and Logarithmic Equations Solving an Exponen.al Equa.on Common Base Any equa+on that contains the form b cx, such as a = b cx, where the exponent includes a variable, is an exponen+al equa+on. If possible, rewrite each side as an expression of a common base raised to an exponent. Examples: What is the solu+on of 16 3x = 8? 16 3x = 8 (2 4 ) 3x = x = x = 3 x = ¼ What is the solu+on of 27 3x = 81? 27 3x = 81 (3 3 ) 3x = x = 3 4 9x = 4 x = 4 / 9

25 7.5 Exponential and Logarithmic Equations Solving an Exponen.al Equa.on Different Bases When the bases are not the same, you can take the logarithm of each side of the equa+on. If m and n are posi+ve, and m = n, then log m = log n. Examples: What is the solu+on of 15 3x = 285? 15 3x = 285 log15 3x = log285 (3x)log15 = log285 x What is the solu+on of 5 2x = 130? 5 2x = 130 ln5 2x = log130 (2x)ln5 = log130 x

26 7.5 Exponential and Logarithmic Equations Solving an Exponen.al Equa.on With a Graph What is the solu+on for 4 3x = 6000? Graph each half of the equa+on separately: Y 1 = 4 3x Y 2 = 6000 Adjust the window to view the point of intersec+on. The solu+on is x 2.09 What is the solu+on for 7 4x = 800? Y 1 = 7 4x Y 2 = 800 The solu+on is x

27 7.5 Exponential and Logarithmic Equations Modeling with an Exponen.al Func.on Wood is a sustainable, renewable, natural resource when you manage forests properly. Your lumber company has 1,200,000 trees. You plan to harvest 7% of the trees each year. How many years will it take to harvest half of the trees? T(n) = a = r = 7% = 0.07 b = 1 + r = 1 + ( 0.07) = 0.93 n =? It will take about 9.55 years to harvest half of the original trees. T(n) = ab n = (0.93) n 0.5 = 0.93 n ln(0.5) = ln(0.93 n ) ln(0.5) =n ln(0.93) 9.55 n

28 7.5 Exponential and Logarithmic Equations Solving a Logarithmic Equa.on A logarithmic equa+on is an equa+on that includes one or more logarithms involving a variable. Examples: What is the solu+on of log(4x 3) = 2? log(4x 3) = 2 10 log(4x 3) = x 3 = 100 4x = 103 x = 103/4 = What is the solu+on of ln (3 2x) = 1? ln (3 2x) = 1 e ln(3 2x) = e 1 3 2x = e 1 2x = e 1 3 x = (e 1 3)/

29 7.5 Exponential and Logarithmic Equations Using Logarithmic Proper.es to Solve an Equa.on What is the solu+on of log (x 3) + log (x) = 1? log(x 3) + log(x) = 1 log ((x 3)x) = 1 log (x 2 3x) = 1 10 log(x2 3x) = 10 1 x 2 3x = 10 x 2 3x 10 = 0 (x 5) (x + 2) = 0 x = 2, 5 Check:? log( 2 3) + log( 2) = 1? log(5 3) + log(5) = 1? log(2) + log(5) = = 1 x = 5

30 7.6 Natural Logarithms Simplifying a Natural Logarithmic Expression The func+on y = e x has an inverse, the natural logarithmic func+on y = log e x, or y = ln x. Example: What is 2ln 15 ln 17 wriuen as a single natural logarithm? 2ln 15 ln 17 = ln 15 2 ln 17 = ln 3

31 7.6 Natural Logarithms Solving a Natural Logarithmic Equa.on Use the inverse rela+onship between the func+ons y = ln x and y = e x to solve certain logarithmic and exponen+al equa+ons. Example: What are the solu+ons of ln (x 3) 2 = 4? ln (x 3) 2 = 4 e ln(x 3)2 = e 4 (x 3) 2 = e 4 x 3 = ±e 2 x = 3 ± e 2 x 10.39, 4.39 Check: ln ( ) 2? = 4 ln ( ) 2? =

32 7.6 Natural Logarithms Solving an Exponen.al Equa.on What is the solu+on of 4e 2x + 2 = 16? 4e 2x + 2 = 16 4e 2x = 14 e 2x = 3.5 ln e 2x = ln 3.5 2x lne = ln 3.5 x = (ln 3.5) / 2 x What is the solu+on of e 3x + 5 = 15? e 3x + 5 = 15 e 3x = 10 ln e 3x = ln 10 3x lne = ln 10 x = (ln 10) / 3 x 0.77

33 7.6 Natural Logarithms Using Natural Logarithms A spacecray can auain a stable orbit 300km above Earth if it reaches a velocity of 7.7 km/s. The formula for a rocket s maximum velocity v in kilometers per second is v = t + c lnr. The booster rocket fires for t seconds and the velocity of the exhaust is c km/s. The ra+o of the mass of the rocket filled with fuel to its mass without fuel is R. Suppose a rocket with exhaust velocity 2.8km/s has a mass ra+o of 25, and a firing +me of 100s. Can the spacecray auain a stable orbit 300km above Earth? R = 25, c = 2.8, t = 100 v = (100) + (2.8) ln(25) v (3.219) v 8.0 km/s Since 8.0km/s > 7.7km/s, the rocket can auain a stable orbit 300km above Earth