lesson 8.1 n practice a ame...

14 Algebra 2Chapter 8 Resource Book

Copyright © McDougal Littell Inc. All rights reserved.

Practice AFor use with pages 465–472

8.1LESSON

NAME _________________________________________________________ DATE ___________

Less

on

8.1

Match the function with its graph.

1. 2. 3.

4. 5. 6.

A. B. C.

D. E. F.

Explain how the graph of g can be obtained from the graph of f.

7. 8. 9.

10. 11. 12.

Identify the y-intercept and asymptote of the graph of the function.

13. 14. 15.

16. 17. 18.

19. Account Balance You deposit $2000 in an account that earns 5% annualinterest. Find the balance after 1 year if the interest is compounded withthe given frequency.

a. annually b. quarterly c. monthly

y ! "14

!4x"y ! "2!4x"y !12

!4x"

y ! 2!4x"y ! #65$

x

y ! 3x

g !x"! 3!2x" # 2g !x"! "2xg !x"! 5x"3

f !x"! 3!2x"f !x"! 2xf !x"! 5x

g !x"! #53$

x#1g !x"! 2x " 5g !x"! #4

3$x

# 2

f !x"! #53$

x

f !x"! 2xf !x"! #43$

x

y

x1

1

(1, 3)

(0, 1)

y

1

1(0, )12 (1, )3

2

x

y

x1

"2

(1, "6)

(0, "2)

y

1

1

(0, " )12 (1, " )3

2

x

y

x1

1

(1, "3)(0, "1)

y

x1

2

(1, 6)

(0, 2)

f !x"! "2!3x"f !x"! "12

!3x"f !x"!12

!3x"

f !x"! 2!3x"f !x"! "3xf !x"! 3x

Algebra 2 15Chapter 8 Resource Book


Practice BFor use with pages 465–472

8.1LESSON

NAME _________________________________________________________ DATE ___________Lesso

n 8

.1NAME _________________________________________________________ DATE ___________


1. 2. 3.

4. 5. 6.

A. B. C.

D. E. F.

Explain how the graph of g can be obtained from the graph of f.

7. 8. 9.

Identify the y-intercept and the asymptote of the graph of the function.

10. 11. 12.

Graph the function.

13. 14. 15.

16. 17. 18.

19. 20. 21.

Computer Usage In Exercises 22–24, use the following information.

From 1991 through 1995, the number of computers per 100 people worldwidecan be modeled by where t is the number of years since 1991.

22. Identify the initial amount, the growth factor, and the annual percentincrease.

23. Graph the function.

24. Estimate the number of computers per 1000 people worldwide in 2000.

C ! 25.2!1.15"t

y ! #32$

x"2

# 1y ! 3x#2 " 1y ! 2x"1 # 4

y ! 2x#1 # 3y ! 3x"1 " 2y ! 2x # 3

y ! 3x " 1y ! 2x#3y ! 4x"2

y ! 3x"1 # 2y ! 3x#3y ! 3x " 2

g !x"! 3x"2 # 4g !x"! #10x"2g !x"! #12$

x#1

"2

f !x"! 3x

f !x"! 10xf !x"! #12$

x

y

x1

1

(2, #3)

(0, # )13

y

x1

1

(2, 3)

(0, )13

y

x1

1

(0, #1)

(#1, 1)

y

x1

1

(#2, 0)

(0, )79

y

x1

1

(0, 3)(#1, 2 )3

4

y

x1

1

(#1, 3)(0, 3 )1

3

f !x"! #3x"1 " 2f !x"! #3x#1f !x"! 3x#1

f !x"! #43$

x"2

# 1f !x"! #43$

x"1

" 2f !x"! #43$

x

" 2




8.2LESSON

NAME _________________________________________________________ DATE ___________Lesso

n 8

.2

Tell whether the function represents exponential growth orexponential decay.

1. 2. 3.

4. 5. 6.


7. 8. 9.

10. 11. 12.

A. B. C.

D. E. F.

Identify the y-intercept and asymptote of the graph of the function.

13. 14. 15.

16. 17. 18.

19. Radioactive Decay Ten grams of Carbon 14 is stored in a container. Theamount C (in grams) of Carbon 14 present after t years can be modeled by

How much Carbon 14 is present after 1000 years?C ! 10!0.99987"t.

y ! "23

#15$

x

y ! "5#12$

x

y !14

#89$

x

y ! 2#13$

x

y ! !0.3"xy ! #23$

x

y

x1

2

(1, )16(0, )1

2

y

x2

1

(0, "1)

(1, " )13

y

x1

1(0, 2) (1, )2

3

y

x3

1

(0, "2) (1, " )23

y

x2

1

(1, " )16

(0, " )12

y

x1

3

(0, 1) (1, )13

y ! "12#1

3$x

y ! "2#13$

x

y !12#1

3$x

y ! 2#13$

x

y ! "#13$

x

y ! #13$

x

f !x"!12

!3x"f !x"! #13$

x

f !x"! !0.7"x

f !x"! 6xf !x"! #54$

x

f !x"! #23$

x




8.2LESSON

NAME _________________________________________________________ DATE ___________

Less

on

8.2

Tell whether the function represents exponential growth orexponential decay.

1. 2. 3.


4. 5. 6.

7. 8. 9.

A. B. C.

D. E. F.

Graph the function.

10. 11. 12.

13. 14. 15.

Value of the Dollar In Exercises 16–18, use the following information.

From 1990 through 1998, the value of the dollar has been shrinking. That is,you cannot buy as much with a dollar today as you could in 1990. Theshrinking value can be modeled by where t is the number ofyears since 1990.

16. How much was a 1998 dollar worth in 1993?

17. Graph the model.

18. Estimate the year in which the 1998 dollar was worth $1.07.

V ! 1.24!0.973"t,

y ! #12$

x"2

" 1y ! #35$

x#1

" 3y ! 2#15$

x"3

y ! 3#14$

x#1

y ! 2#13$

x

# 4y ! 2#12$

x

" 3

y

x1

2

(0, 5)

(#1, 13)y

x1

1

(#1, 5)

(0, 1)

y

x1

1

(#2, 2)

(0, # )2225

y

x1

25(3, 1)

(0, 125)y

x2

1

(0, #1)

(#1, #5)

y

x1

1(#2, 1) (0, )1

25

y ! ##15$

x

y ! 3#15$

x"2

# 1y ! 2#15$

x

" 3

y ! #15$

x#3

y ! #15$

x"2

y ! #15$

x

f !x"! 3!4"#xf !x"!13#7

5$x

f !x"!12#5

7$x




8.3LESSON

NAME _________________________________________________________ DATE ___________

Use a calculator to evaluate the expression. Round the result tothree decimal places.

1. 2. 3. 4.

Tell whether the function is an example of exponential growth orexponential decay.

5. 6. 7.

8. 9. 10.

Simplify the expression.

11. 12. 13.

14. 15. 16.

17. 18. 19.

Complete the table of values. Round to two decimal places.

20. 21.

22. 23.

Graph the function and identify the horizontal asymptote.

24. 25. 26.

27. 28. 29.

Interest In Exercises 30–32, use the following information.

You deposit $1200 in an account that pays 5% annual interest. After 10 years,you withdraw the money.

30. Find the balance in the account if the interest was compounded quarterly.

31. Find the balance in the account if the interest was compoundedcontinuously.

32. Which type of compounding yielded the greatest balance?

f !x" ! e"2.5x " 3f !x" ! 12 e2x " 1f !x" ! e"3x # 1

f !x" ! ex # 2f !x" ! 2e"xf !x" ! 2ex

f !x" ! e"3x " 2f !x" ! e2x # 3

f !x" ! 2e"xf !x" ! 2ex

eex#1e2x $ e1"2x#64e4x

2ex $ ex#3"3e $ 4e2!4e3"2

$e2%

"13e5

e!e4""2

f !x" ! 4e5xf !x" ! 12 e"xf !x" ! 1

5 e5x

f !x" ! 2e"3xf !x" ! e"3xf !x" ! 2e3x

e#2e"1.4e"1&3e5

Less

on

8.3

x 0 1 1.5 2f(x)

"1"1.5"2

x 0 1 1.5 2f(x)

"1"1.5"2 x 0 1 1.5 2f(x)

"1"1.5"2

x 0 1 1.5 2f(x)

"1"1.5"2



Practice CFor use with pages 480–485

8.3LESSON

NAME _________________________________________________________ DATE ___________


1. 2. 3. 4.


5. 6. 7.

8. 9. 10.

Identify the horizontal asymptote of the function.

11. 12. 13.

Graph the function. State the domain and range.

14. 15. 16.

17. 18. 19.

Carbon Dating In Exercises 20–22, use the following information.

Carbon dating is a process to estimate the age of organic material. In carbondating the formula used is

where R is the ratio of Carbon 14 to Carbon 12 and t is time in years.

20. Is the model an example of exponential growth or exponential decay?


22. Use the graph to estimate the age of a fossil whose Carbon 14 to Carbon12 ratio is

Learning Curve In Exercises 23–26, use the following information.

The management at a factory has determined that a worker can produce a maxi-mum of 30 units per day. The model indicates the numberof units y that a new employee can produce per day after t days onthe job.

23. Is the model an example of exponential growth or exponential decay?


25. How many units can be produced per day by an employee who has beenon the job 8 days?

26. Use the graph to estimate how many days of employment are required fora worker to produce 25 units per day.

y ! 30 " 30e"0.07t

3 # 10"13.

R !1

1012 e"t!8233

f "x# ! 54 e2"x"1# " 3f "x# ! 2

3 e3"x $ 1f "x# ! 12 e2x$1 " 5

f "x# ! 2ex"4 $ 1f "x# ! 14 e"x " 2f "x# ! 2e3x $ 1

f "x# ! 245e"0.023xf "x# ! 12 e3x$1 $ 4f "x# ! 3e2x " 1

3$8e12x%e3x

2e&2

"4e1%2x#"6

%e2

2 &"3

%13

e"2&"4

e2"2e4#3

eee" 1$2e"2.6e$3

Lesson

8.3




8.4LESSON

NAME _________________________________________________________ DATE ___________

Rewrite the equation in exponential form.

1. 2. 3.

4. 5. 6.

Evaluate the expression without using a calculator.

7. 8. 9.

10. 11. 12.


13. 14. 15.

16. 17. 18.


19. 20. 21.

22. 23. 24.


25. 26. 27.

A. B. C.

Match the function with the graph of its inverse.

28. 29. 30.

A. B. C.

31. Sound The level of sound V in decibels with an intensity I can be modeled by

where I is intensity in watts per centimeter. Loud music can have an intensity of watts percentimeter. Find the level of sound of loud music.

10!5

V " 10 log! I10!16",

y

x1

1

y

x1

2

y

x1

1

f #x$ " ln xf #x$ " log1%3 xf #x$ " log x

y

x1

1

y

x1

1

y

x1

1

f #x$ " log1%2 xf #x$ " log5 xf #x$ " log3 x

log221#221x$log15#15x$log3#3x$13log13 x27log27 x7log7 x

ln #6.12$ln #0.23$ln 8

log 3.72log #0.4$log 6

log8 8log7 1log10 100


log6 6 " 1log2 16 " 4log7 49 " 2

log3 27 " 3log5 25 " 2log2 8 " 3

Less

on

8.4




8.4LESSON

NAME _________________________________________________________ DATE ___________

Rewrite the equation in exponential form.

1. 2. 3.

4. 5. 6.


7. 8. 9.

Evaluate the logarithm without using a calculator.

10. 11. 12.

13. 14. 15.

Find the inverse of the function.

16. 17. 18.

19. 20. 21.

Graph the function.

22. 23. 24.

25. 26. 27.

28. Galloping Speed Four-legged animals run with two different types ofmotion: trotting and galloping. An animal that is trotting has at least onefoot on the ground at all times. An animal that is galloping has all fourfeet off the ground at times. The number s of strides per minute at whichan animal breaks from a trot to a gallop is related to the animal’s weight w(in pounds) by the model

Approximate the number of strides per minute for a 500 pound horsewhen it breaks from a trot to a gallop.

29. Tornadoes The wind speed S (in miles per hour) near the center ofa tornado is related to the distance d (in miles) the tornado travels bythe model

Approximate the wind speed of a tornado that traveled 150 miles.

S ! 93 log d " 65.

S ! 256.2 # 47.9 log w.

f !x" ! #1 " log6 xf !x" ! log6 !2x"f !x" ! #log6 x

f !x" ! log6 !x " 1"f !x" ! 1 " log6 xf !x" ! log6 x

f !x" ! log4 16xf !x" ! log2 !x # 1"f !x" ! log 2x

f !x" ! log1#3 xf !x" ! ln xf !x" ! log3 x

log6 !#1"log5 52#3log8 2


ln !23"log 11.5ln $3

log2 18 ! #3log5

15 ! #1log9 3 ! 1

2

log2 1 ! 0log3 81 ! 4log4 16 ! 2

Lesson

8.4




8.5LESSON

NAME _________________________________________________________ DATE ___________

Use the properties of logarithms to rewrite the expression in termsof and Then use and toapproximate the expression.

1. 2. 3.

4. 5. 6.

Expand the expression.

7. 8. 9.

10. 11. 12.

13. 14. 15.

Condense the expression.

16. 17. 18.

19. 20. 21.

22. 23. 24.

Use the change-of-base formula to rewrite the expression. Thenuse a calculator to evaluate the expression. Round your result tothree decimal places.

25. 26. 27.

28. 29. 30.

Investments In Exercises 31 and 32, use the following information.

You want to invest in a stock whose value has been increasing by approximately5% each year. The time required for an initial investment of to grow to I canbe modeled by

where and I are measured in dollars and t is measured in years.

31. Expand the expression for t.

32. Assume that you have $1000 to invest. Complete the table to show howlong your investment would take to double, triple, and quadruple.

I0

t !ln! I

I0"

0.049,

I0

log4 1235log5 12log6 200


2 log x " log 8log3 #x " 5$ " log3 4ln 2 # ln #x " 2$log #x # 1$ # log 6ln x # ln 3log 4 # log x

log3 14 " log3 ylog2 x " log2 7log 3 " log 5

log3#27x$2log2%2xlog 3%x

ln x#3log3 x5log6!6x"

log!x5"log3#9x$log2#3x$

log 49log 7#3log #27$

log #72$log 14log 4

log 7 ≈ 0.845log 2 ≈ 0.301log 7.log 2

Less

on

8.5

I 2000 3000 4000t




8.5LESSON

NAME _________________________________________________________ DATE ___________

Use the properties of logarithms to rewrite the expression in termsof and Then use and toapproximate the expression.

1. 2. 3.

4. 5. 6.


7. 8. 9.

10. 11. 12.

13. 14. 15.


16. 17.

18. 19.

20. 21.

Use the change-of-base formula to rewrite the expression. Thenuse a calculator to evaluate the expression. Round your result tothree decimal places if necessary.

22. 23. 24.

25. 26. 27.

Henderson-Hasselbach Formula In Exercises 28–32, use the followinginformation.

The pH of a patient’s blood can be calculated using the Henderson-HasselbachFormula, where B is the concentration of bicarbonate and C isthe concentration of carbonic acid. The normal pH of blood is approximately 7.4.

28. Expand the right side of the formula.

29. A patient has a bicarbonate concentration of 24 and a carbonic acidconcentration of 1.9. Find the pH of the patient’s blood.

30. Is the patient’s pH in Exercise 29 below normal or above normal?

31. A patient has a bicarbonate concentration of 24. Graph the model.

32. Use the graph to approximate the concentration of carbonic acid requiredfor the patient to have normal blood pH.

pH ! 6.1 " log BC ,

log1!2 6log1.5 2.8log0.8 12

log4 0.5log6 2log3 12

log3 4 " 2 log3 x # log3 523 log2 x # 3 log2 y

12 log x # log 4log4 5 " log4 x " log4 y

2 log5 x " log5 3log3 7 # log3 x

log2 x2yz

log 10"x

log x2

4

log5 2"xlog3 "x y zlog4 xy3

log xy2log2 x5

log6 3x

log # 427$log 14log 16

log 9log 12log #34$

log 4 ≈ 0.602log 3 ≈ 0.477log 4.log 3

Lesson

8.5




8.6LESSON

NAME _________________________________________________________ DATE ___________Lesso

n 8

.6

Tell whether the x-value is a solution of the equation.

1. 2. 3.

4. 5. 6.

Tell whether the x-value is a solution of the equation.

7. 8. 9.

10. 11. 12.

Solve the equation.

13. 14. 15.

16. 17. 18.

Solve the equation by taking the appropriate log of each side.

19. 20. 21.

22. 23. 24.

Use the following property to solve the equation. For positivenumbers b, x, and y where if and only if

25. 26. 27.

28. 29. 30.

Solve the equation by exponentiating each side.

31. 32. 33.

34. 35. 36.

Compound Interest You deposit $100 in an account that earns 3%annual interest compounded continuously. How long does it takethe balance to reach the following amounts?

37. $110 38. $150 39. $200

log!4x" ! 1 " 3ln!3x ! 1" " 0ln!5x # 3" " 2

log!2x ! 3" " 6log3!x # 1" " 8log2 x " 5

log!3x ! 2" " log!x # 1"ln!x ! 3" " ln!6 # 3x"log3!x # 1" " log3!2x ! 5"log2!4x" " log2 12log!x ! 2" " log 9log x " log 7

x ! y.logb x ! logb yb " 1,

53x # 2 " 82x ! 5 " 12e2x " 6

ex " 53x " 102x " 9

10x " 107#3xe2x#1 " e3#xe3x " e2x!7

24x!1 " 22x#332x " 3x#54x " 42x!1

5ex ! 2 " 17, x " ln 33ex # 1 " 11, x " 42ex " 8, x " ln 4

ex " 3, x " log 3ex " 7, x " ln 7ex " 5, x " 5

ln 2x " 14, x " 2e14ln 6x " 4, x "e4

6ln 2x " 8, x " e8

ln x " 7, x " 7eln x " 3, x " 3eln x " 9, x " e9




8.6LESSON

NAME _________________________________________________________ DATE ___________

Less

on

8.6

Solve the exponential equation. Round the result to three decimal places if necessary.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21.

22. 23. 24.

25. 26. 27.

Solve the logarithmic equation. Round the result to three decimal places if necessary.

28. 29. 30.

31. 32. 33.

34. 35. 36.

37. 38. 39.

40. 41. 42.

43. 44. 45.

46. 47. 48.

49. Compound Interest You deposit $2000 into an account that pays 2%annual interest compounded quarterly. How long will it take for the balance to reach $2500?

50. Rocket Velocity Disregarding the force of gravity, the maximumvelocity v of a rocket is given by where t is the velocity of the exhaust and M is the ratio of the mass of the rocket with fuel to itsmass without fuel. A solid propellant rocket has an exhaust velocity of 2.5 kilometers per second. Its maximum velocity is 7.5 kilometers persecond. Find its mass ratio M.

v ! t ln M,

ln !4x " 9" ! ln xln !2x # 3" ! ln !2x " 1"ln !5x " 1" ! ln !3x # 2"log3 !2x # 1" ! log3 !x " 4"log2 !x # 2" ! log2 3x2 # log2 3x ! 8

ln 4x " 6 ! 82 # log3 2x ! "3log2 5x ! 1

log3 3x ! 29 log10 x " 4 ! 113 log10 x # 1 ! 13

"5 # 2 ln x ! 54 " ln x ! 1"3 # ln x ! 5

7 # log10 x ! 42 log10 x ! 107 ln x ! 21

log2 x ! 1.5log10 x ! "2ln x ! 5

38!23x" # 1 ! 102

3e2x ! 1213ex # 1 ! 5

20.1x # 6 ! 12"3e"x " 4 ! "13"4e2x # 3 ! "5

2!23x" ! 23e5x ! 142e4x ! 5

3e"x ! 184!2x" ! 162ex ! 10

e"x " 6 ! 1e"2x # 5 ! 124"2x " 3 ! 1

2"x # 1 ! 6e4x " 3 ! 7e3x # 6 ! 10

32x " 3 ! 4e2x ! 523x ! 4

52x ! 82x # 7 ! 10ex # 3 ! 8

e2x ! 4210x ! 350ex ! 18




8.7LESSON

NAME _________________________________________________________ DATE ___________

Write an exponential function of the form whose graphpasses through the given points.

1. 2. 3.

4. 5. 6.

Use the table of values to determine whether or not an exponen-tial model is a good fit for the data

7.

8.

9.

10.

Solve for

11. 12. 13.

14. 15. 16.

Write a power function of the form whose graph passesthrough the given points.

17. 18. 19.

Use the table of values to determine whether or not a powerfunction model is a good fit for the data

20.

21.

Solve for

22. 23. 24. ln y ! 0.8 ln xln y ! 1.3 ln xln y ! 2.4 ln x

y.

!x, y".

(1, 1#, $4, 8#$1, 3#, $2, 12#$1, 2#, $3, 54#

y ! axb

ln y ! 2.301t " 1.624ln y ! 1.032t " 8.149ln y ! 3.207t " 1.132

ln y ! 12.135t " 5.144ln y ! 1.203t " 0.418ln y ! 0.324t " 1.601

y.

!t, y".

$2, 18#, $3, 54#$4, 16#, $6, 64#$1, 2#, $2, 8#$1, 10#, $2, 50#$1, 6#, $2, 12#$0, 1), $3, 27#

y ! abx

Lesson

8.7

t 1 2 3 4 5 6 7 8ln y 0.23 0.64 1.07 1.47 1.88 2.31 2.72 3.12

t 1 2 3 4 5 6 7 8ln y 1.32 1.52 1.92 2.72 2.88 3.52 4.32 5.6

t 1 2 3 4 5 6 7 8ln y 0.05 0.17 0.27 0.40 0.52 0.63 0.75 0.85

t 1 2 3 4 5 6 7 8ln y 12.31 13.56 14.82 16.04 17.29 18.49 19.76 21.01

ln x 0 0.693 1.099 1.386 1.609ln y 1.264 2.594 3.924 5.254 6.584

ln x 0 0.693 1.099 1.386 1.609ln y 0.833 2.219 3.030 3.605 4.052




8.7LESSON

NAME _________________________________________________________ DATE ___________

Write an exponential function of the form whose graphpasses through the given points.

1. 2. 3.

4. 5. 6.

Use the table of values to draw a scatter plot of versus Thenfind an exponential model for the data.

7.

8.

9.

Write a power function of the form whose graph passesthrough the given points.

10. 11. 12.

13. 14. 15.

Use the table of values to draw a scatter plot of versus Then find a power model for the data.

16.

17.

18. Consumer Magazines The table shows the circulation of the top 10 con-sumer magazines in 1997 where x represents the magazine’s ranking. Usea graphing calculator to find a power model for the data. Use the model toestimate the circulation of the 15th ranked magazine.

ln x.ln y

!2, 9.879", !3, 16.070"!4, 384", !16, 49,152"!4, 9.6", !9, 32.4"!2, 64", !3, 486"!2, 4", !4, 32"!2, 16", !3, 36"

y ! axb

x.ln y

#2, 536$, #3,

5108$#1,

52$, #2,

252 $!1, 4", #2,

83$

#2, 34$, #3,

38$#2,

1625$, #3,

6425$#1,

23$, #2,

43$

y ! abx

Less

on

8.7

x 1 2 3 4 5 6 7 8y 8 16 32 64 128 256 512 1024

x 1 2 3 4 5 6 7 8y 3.6 8.64 20.736 49.766 119.439 286.654 687.971 1651.13

x 1 2 3 4 5 6 7 8y 3 4.5 6.75 10.125 15.188 22.781 34.172 51.258

x 1 2 3 4 5 6 7 8y 1.5 6 13.5 24 37.5 54 73.5 96

x 1 2 3 4 5 6 7 8y 2.4 7.275 13.919 22.055 31.518 42.194 53.997 66.858

Rank Circulation Rank Circulation(millions) (millions)

1 20.454 6 7.6152 20.432 7 5.0543 15.086 8 4.6434 13.171 9 4.5145 9.013 10 4.256




8.8LESSON

NAME _________________________________________________________ DATE ___________

Evaluate the function for the given value of

1. 2. 3. 4.

5. 6. 7. 8.


9. 10. 11.

A. B. C.

Identify the horizontal asymptotes of the function.

12. 13. 14.

Identify the y-intercept of the function.

15. 16. 17.

Identify the point of maximum growth of the function.

18. 19. 20.

Advertising In Exercises 21 and 22, use the following information.

A company decides to stop advertising one of its products. The sales of theproduct S can be modeled by

where t is the number of years since advertising stopped.

21. What are the sales 5 years after advertising stopped?

22. What can the company expect in terms of sales in the future?

S !100,000

1 " 0.5e#0.3t

f !x" !2

1 " 2e#3xf !x" !1

1 " 3e#xf !x" !4

1 " e#2x

y !5

1 " e#3xy !4

1 " e#xy !1

1 " 2e#x

f !x" !6

1 " 2e#xf !x" !5

1 " e#2xf !x" !1

1 " 4e#2x

f !x" !1

1 " 2e#xf !x" !3

1 " e#2xf !x" !3

1 " e#x

f !54"f !#0.2"f !3.4"f !1

2"f !0"f !6"f !#1"f !1"

x.f #x$ !2

1 " e#x

Less

on

8.8




8.8LESSON

NAME _________________________________________________________ DATE ___________

Tell whether the function is an example of exponential growth,exponential decay, logarithmic, or logistics growth.

1. 2. 3.

4. 5. 6.


7. 8. 9.

A. B. C.

Identify the horizontal asymptotes of the function.

10. 11. 12.

Sketch the graph of the function.

13. 14. 15.

Solve the equation.

16. 17. 18.

Wildlife Management In Exercises 19–22, use the following information.

A wildlife organization releases 100 deer into a wilderness area. The deerpopulation P can be modeled by

where t is the time in years.

19. Sketch the graph of the model.

20. Identify the horizontal asymptotes of the graph.

21. What is the maximum number of deer the wilderness area can support?

22. What is the deer population after 10 years?

P !500

1 " 4e#0.36t

121 " 5e#2x ! 6

81 " e#x ! 5

41 " 2e#x ! 2

f !x" ! 1 "5

1 " e#xf !x" !1

1 " 5e#xf !x" !3

1 " e#x

f !x" ! 10 "2

1 " e#xf !x" ! #5 "1

1 " e#xf !x" !20

1 " 0.4e#x

f !x" !4

1 " e#2xf !x" !2

1 " 2e#xf !x" !4

1 " 2e#x

f !x" ! log 6xf !x" ! 2.5xf !x" ! e#2x

f !x" !1

1 " 3e#xf !x" ! ln 3xf !x" ! #12$

x

Lesson

8.8


8 SAT/ACT Chapter TestFor use after Chapter 8

NAME _________________________________________________________ DATE ____________


Review

and

Assess

CHAPTER

1. What is the log of 100 to the base 10?

A 10 B 2

C 3 D 1

2. The natural base e is

A rational B imaginary

C irrational D undefined

3. What is the simplified form of

A B

C D

4. What type of function is

A Exponential deacy function

B Linear function

C Quadratic function

D Exponential growth

5. Which of the following is equivalent to

A B 125

C 3 D

6. What is the solution of the equation

A No solution B

C 5 D 2

7. What is the asymptote of the graph of

A x-axis B y-axis

C D y ! "1y ! 1

f !x" ! 2x?

"2

9x#1 ! 27x"1?

"125

"3

log553?

f !x" ! 2e2x?

"e2

12e

"2e"2e7

8e4

"4e3?

8. Which of the following is equivalent to

A B

C D

Quantitative Comparision Exercises 9 and 10,choose the statement that is true about the givenquantities.

A The quantity in column A is greater.B The quantity in column B is greater.C The two quantities are equal.D The relationship cannot be determined

from the given information.

9.

10.

logbx # logbylogb!x " y"1#2

logbx " logbylogb x $ logby

logb xy?

Column A Column Blog101000 log327

Column A Column BThe y value when

of the graphof y ! "2 % 4xx ! 0

The y value whenof the graph

of y ! 2 % 5xx ! 0

Chapter ReviewCHAPTER

8

37A Chapter 8 Algebra 2 English-Spanish Reviews

VOCABULARY

8.1 EXPONENTIAL GROWTH

An exponential growth function has the form y = abx with a > 0 and b > 1.

To graph y = 2 • 5x + 2 º 4, first lightly sketch the graph of y = 2 • 5x, which passes through (0, 2) and (1, 10). Then translate the graph 2 units to the left and 4 units down. The graph passes through (º2, º2) and (º1, 6). The asymptote is the line y = º4. The domain is all real numbers,and the range is y > º4.

EXAMPLE

Examples onpp. 465–468


1. y = º2x + 4 2. y = 3 • 2x 3. y = 5 • 3x º 2 4. y = 4x + 3 º 1

• exponential function, p. 465

• base of an exponential function, p. 465

• asymptote, p. 465

• exponential growth function, p. 466

• growth factor, p. 467

• exponential decay function, p. 474

• decay factor, p. 476

• natural base e, or Euler number, p. 480

• logarithm of y with base b, p. 486

• common logarithm, p. 487

• natural logarithm, p. 487

• change-of-base formula, p. 494

• logistic growth function, p. 517

8.2 EXPONENTIAL DECAY

An exponential decay function has the form y = abx witha > 0 and 0 < b < 1.

To graph y = 4!!13!"x

, plot (0, 4) and !1, !43!". From right to left draw a

curve that begins just above the x-axis, passes through the two points, and moves up. The asymptote is the line y = 0. The domain is all realnumbers, and the range is y > 0.

EXAMPLE


Tell whether the function represents exponential growth or exponential decay.

5. ƒ(x) = 5!!34!"x

6. ƒ(x) = 2!!54!"x

7. ƒ(x) = 3(6)ºx 8. ƒ(x) = 4(3)x


9. y = !!14!"x

10. y = 2!!35!"x º 1

11. y = !!12!"x

º 5 12. y = º3!!34!"x

+ 2

Copyright © McDougal Littell Inc.All rights reserved.

2

"4

x

y ! 2 ! 5x

y ! 2 ! 5 x " 2 # 4

x1

1

y ! 4# $x13

y



13. y = ex + 5 14. y = 0.4ex º 3 15. y = 4eº2x 16. y = ºex + 3

Evaluate the expression without using a calculator.

17. log4 64 18. log2 !18! 19. log3 !

19! 20. log6 1


21. y = 3 log5 x 22. y = log 4x 23. y = ln x + 4 24. y = log (x º 2)

8.4 LOGARITHMIC FUNCTIONS

You can use the definition of logarithm to evaluate expressions: logb y = x if and only if bx = y. The common logarithm has base 10 (log10 x = log x).The natural logarithm has base e (loge x = ln x).

To evaluate log8 4096, write log8 4096 = log8 84 = 4.

To graph the logarithmic function ƒ(x) = 2 log x + 1, plot points such as (1, 1) and (10, 3). The vertical line x = 0 is an asymptote. The domain is x > 0, and the range is all real numbers.

EXAMPLES


8.3 THE NUMBER e

You can use e as the base of an exponential function.To graph such a function, use e ≈ 2.718 and plot some points.

ƒ(x) = 3e2x is an exponential growth function, since 2 > 0.g(x) = 3eº2x is an exponential decay function, since º2 < 0.

For both functions, the y-intercept is 3, the asymptote is y = 0, the domain is all real numbers, and the range is y > 0.

EXAMPLES



25. log3 6xy 26. ln !73x! 27. log 5x3 28. log !

x5

2yy

º2!


29. 2 ln 3 º ln 5 30. log4 3 + 3 log4 2 31. 0.5 log 4 + 2(log 6 º log 2)

8.5 PROPERTIES OF LOGARITHMS

You can use product, quotient, and power properties of logarithms.

Expand: log2 !3yx! = log2 3x º log2 y = log2 3 + log2 x º log2 y

Condense: 3 log6 4 + log6 2 = log6 43 + log6 2 = log6 (64 • 2) = log6 128

EXAMPLES



x1

3y ! 3e 2x y ! 3e"2x

y

x1

1 ƒ( x) ! 2 log x # 1


8.6 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS

Solve the equation. Check for extraneous solutions.

32. 2(3)2x = 5 33. 3eºx º 4 = 9 34. 3 + ln x = 8 35. 5 log (x º 2) = 11

You can solve exponential equations by equating exponents or by takingthe logarithm of each side. You can solve logarithmic equations by exponentiating eachside of the equation.

10x = 4.3 log4 x = 3

log 10x = log 4.3 Take log of each side. 4log4 x = 43 Exponentiate each side.

x = log 4.3 ≈ 0.633 x = 43 = 64


EXAMPLES


8.7 MODELING WITH EXPONENTIAL AND POWER FUNCTIONS

Find an exponential function of the form y = abx whose graph passes throughthe given points.

36. (2, 6), (3, 8) 37. (2, 8.9), (4, 20) 38. (2, 4.2), (4, 3.6)

Find a power function of the form y = axb whose graph passes through thegiven points.

39. (2, 3.4), (6, 7.3) 40. (2, 12.5), (4, 33.2) 41. (0.5, 1), (10, 150)

You can write an exponential function of the form y = abx or a powerfunction of the form y = axb that passes through two given points.

To find a power function given (3, 2) and (9, 12), substitute the coordinates into y = axb to get the equations 2 = a • 3b and 12 = a • 9b. Solve the system of equationsby substitution: a ≈ 0.333 and b ≈ 1.631. So, the function is y = 0.333x1.631.


EXAMPLE

8.8 LOGISTIC GROWTH FUNCTIONS

Graph the function. Identify the asymptotes, y-intercept, and point of maximum growth.

42. y = 43. y = 44. y = 3!!1 + 0.5eº0.5x

4!!1 + 2eº3x

2!1 + eº2x

You can graph logistic growth functions by plotting points and identifying important characteristics of the graph.

The graph of y = is shown. It has asymptotes

y = 0 and y = 6. The y-intercept is 1.5. The point of maximum

growth is !!ln

23

!, !62!" ≈ (0.55, 3).

6!!1 + 3eº2x


EXAMPLE y

x1

2(0.55, 3)

lesson 8.1 n practice a ame...

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