Name ______________________ Algebra II Unit 4A Exponential Functions HW #1 Mrs. Dailey 1) For each of the following, determine the integer value of n that satisfies the equation. The first is done for you.

(a)

12

8

n = (b) 3n = 81 (c)

13

9

n =

3

12

2

n =

2n = 2-3 n= -3

(d)

15

125

n = (e)

110

1,000

n = (f) 13n = 1

2) Use the Addition Property of Exponents to simplify each expression. Then, find a final numerical answer without using your calculator. (a) 2-5•23•24 (b) 53•57•5-10 (c) 103•10-7•102 3) Use the Product Property of Exponents to simplify each exponential expression. You do not need to find a final numerical answer.

(a) ( )4

23 (b) ( )2

23− (c) ( )( )

24

23−

−

HW#1

4) The exponential expression

41

8

is not equivalent to which of the following?

(1) 8-4 (2) 2-12 (3) 4

1

8 (4) 32-1

5) Simplify the expression ( )

4 5

23 7

3

2

x y

x y

−

−− and write the answer using only positive

exponents. (June ’11)

6) If a = 3 and b= -2, what is the value of the expression

2

3

a

b

−

− ? (June ’10)

1) 9

8−

3) 8

9−

2) -1 4) 8

9

7) Which expression is equivalent to ( )1

23x−

? (Jan ’14)

1)

2

1

3x

3)

2

1

9x

2) 23x− 4) 29x−

Name ______________________ Algebra II Unit 4A Exponential Functions HW #2 Mrs. Dailey 1) NO CALCULATOR!!!!! Rewrite each of the following as an equivalent root.

(a) 1

236 (b) 1

327 (c) 1

532

(e) 1

4625 (f) 1

249 (g) 1

481−

2) Rewrite each of the following as an equivalent root and evaluate. Resist using a calculator.

a) 2

38 (b) 3

24 (c) 3

416−

(e) 5

24−

(f) 3

7128 (g) 3

4625

3) Given the function 3

2( ) 5( 4)f x x= + , determine the y-intercept, algebraically,

HW#2

4) Which of the following is equivalent to 1

2x−

?

(1) 1

2x− (2) x− (3)

1

x (4)

1

2x−

5) Written without fractional or negative exponents, 3

2x−

is equal to

(1) 3

2

x− (2)

3 2

1

x (3)

3

1

x (4)

1

x−

6) The expression

2

5x−

is equivalent to (Jan ’11)

1) 52 x− 3)

52

1

x

2) 5 2x− 4)

5 2

1

x

7) Which expression is equivalent to ( )1

2 6 29x y−

? (Jan ’13)

1)

3

1

3xy

3)

3

3

xy

2) 3xy3 4) 3

3

xy

8) The expression ( )2

2 31x−

− is equivalent to (June ’10)

1)

3)

2)

4)

9) Marlene claims that the square root of a cube root is a sixth root? Is she correct? To start, try rewriting the expression below in terms of fractional exponents. Then apply the Product Property of Exponents.

3 a

Name ______________________ Algebra II Unit 4A Exponential Functions HW #3 Mrs. Dailey

1) Which of the following equations could describe the graph shown below?

(1) y =x2 + 1 (3) y = -2x +1

(2) 2

3

x

y

=

(4) y = 4x

2) Which of the following equations represents the graph shown?

(1) y =5x (3) 1

22

x

y

= +

(2) y= 4x +1 (4) y = 3x + 2

3) Sketch the graph of 1

183

x

y

=

below on the axes given. Label the y-intercept.

HW#3

4) Which expression is equivalent to ( )1

2 6 29x y−

? (Jan ’13)

1)

3

1

3xy

3)

3

3

xy

2) 3xy3 4) 3

3

xy

5) The expression ( )2

2 31x−

− is equivalent to (June ’10)

1)

3)

2)

4)

Name ______________________ Algebra II Unit 4A Exponential Functions HW #5 Mrs. Dailey

1) Sketch graphs of the equations shown below on the axes given. Label the y-

intercepts of each graph.

(a) y= 25(4)x

2) For each of the following coordinate pairs, find the equation of the exponential

function, in the form y = a(b)x that passes through the pair. Show the work that you use

to arrive at your answer.

(a) (0,180) and (2, 80) (b) (0,10) and (3, 80)

HW#5

3) Engineers are draining a water reservoir until its depth is only 10 feet. The depth decreases exponentially as shown in the graph below. The engineers measure the depth after 1 hour to be 64 feet and after 4 hours to be 28 feet. Develop an exponential equation in y = a(b)x x to predict the depth as a function of hours draining. Round to the nearest hundredth.

4) Write an exponential function of the form y = a(b)x that passes through the points

(2, 3) and (4,27).

Name ______________________ Algebra II Unit 4A Exponential Functions HW #6 Mrs. Dailey 1) Solve each of the following exponential equations using the Method of Common Bases.

(a) 32x-5 = 9 (b) 23x+7=16 (c) 4 5 1

5125

x− =

(d) 8x = 42x+1 (e)

3 2

2 16

216

x

x

−

− =

(f)

15 31

51

312525

xx

+−

=

HW#6

2) Algebraically determine the intersection point of the two exponential functions shown below. y = 8x-1 and y = 42x-3 3) Algebraically determine the zeroes of the exponential function f(x)= 22x-9 - 32 . Recall that the reason it is known as a zero is because the output is zero. 4) Solve for x.

2

51115 10

25

x−

= −

Name ______________________ Algebra II Unit 4A Exponential Functions HW #7 Mrs. Dailey 1) If $130 is invested in a savings account that earns 4% interest per year, which of the following is closest to the amount in the account at the end of 10 years? (1) $218 (3) $168 (2) $192 (4) $324 2) A population of 50 fruit flies is increasing at a rate of 6% per day. Which of the following is closest to the number of days it will take for the fruit fly population to double? (1) 18 (3) 12 (2) 6 (4) 28 3) If a radioactive substance is quickly decaying at a rate of 13% per hour approximately how much of a 200-pound sample remains after one day? (1) 7.1 pounds (3) 25.6 pounds (2) 2.3 pounds (4) 15.6 pounds 4) A population of llamas stranded on a dessert island is decreasing due to a food shortage by 6% per year. If the population of llamas started out at 350, how many are left on the island 10 years later? (1) 257 (3) 102 (2) 58 (4) 189 5) Which of the following equations would model a population with an initial size of 625 that is growing at an annual rate of 8.5%?

(1) P = 625(8.5)t (3) P =1.085t +625 (2) P = 625(1.085)t (4) P =8.5t +625

HW#7

6) The acceleration of an object falling through the air will decrease at a rate of 15% per second due to air resistance. If the initial acceleration due to gravity is 9.8 meters per second per second, which of the following equations best models the acceleration t seconds after the object begins falling? (1) a =15 -9.8t2 (3) a = 9.8(1.15)t

(2) 9.8

1.5a

t= (4) a = 9.8(0.85)t

7) Red Hook has a population of 6,200 people and is growing at a rate of 8% per year. Rhinebeck has a population of 8,750 and is growing at a rate of 6% per year. In how many years, to the nearest year, will Red Hook and Rhinebeck have the same population?

Name ______________________ Algebra II Unit 4A Exponential Functions HW #9 Mrs. Dailey

1) A rabbit population doubles every 4 weeks. There are currently five rabbits in a restricted area. If t represents the time, in weeks, and P(t) is the population of rabbits with respect to time, about how many rabbits will there be in 98 days? 1) 56 2) 152 3) 3,688 4) 81,290 2) A student studying public policy created a model for the population of Detroit, where the population decreased 25% over a decade. He used the model P= 714(0.75)d, where P is the population, in thousands, d decades after 2010. Another student, Suzanne, wants to use a model that would predict the population after y years. Suzanne's model is best represented by 1) P = 714(0.6500)y 2) P = 714(0.8500)y 3) P = 714(0.9716)y 4) P = 714(0.9750)y 3) Last year, the total population for a California town increased 6.35% over the previous year. If this trend were to continue, which expression could be used to approximate their monthly percent increase in population? [Let m represent months.]

1) (1.0635)m

2) ( )12

1.0635 m

3) (1.00514)m

4) ( )121.00514m

HW#9

Name _______________________________ Date _____________ CC Algebra II Unit 4A Linear Functions Mrs. Dailey Topics 1) Recognize that in an exponential function f(x)= abx, “a” is the y-intercept (y-value when x =0), and “b” is the scale factor or multiplier. Questions: 3,7, 10, 13 2) Write an exponential equation given two points. a) Find two points and write two equations. b) Divide the two equations to find “b” c) Plug “b” into one of original equations to find “a.” Questions: 11,18, 25 3) Be able to rewrite expressions with roots using exponents.

( )3 3x

x y y=

Questions: 1,2 15 4) Understand how negative exponents work.

1

2

2

12

2

1 12

4 2

−

−

=

= =

Questions: 6, 16, 22 5) Understand and be able to manipulate expressions using the Laws of Exponents. Questions: 6, 16,22 6) Memorize the following formulas: a) Key Word: Continuously Compound: A = Pert b) Key Word: Compounded Quarterly/Monthly/Daily:

( )A(t)=P 1t

r or ( ) 1

n tr

A t Pn

•

=

Questions: 5,14,21, 26, 28 7) Solving Exponential Equations with different bases Questions: 8,9,23 8) Mindful Manipulations: Given annual rate → get monthly rate Given annual rate → get decade rate Given hourly rate → get daily rate Questions: 4, 12, 20, 27 9) Half-life Questions

( )A(t)=P 1- 0.50t

Half life−

Questions: 17, 24

Review

Name _______________________________ Date _____________ CC Algebra II Unit 4A Linear Functions Mrs. Dailey

1) The expression

5

2a is equivalent to which of the following as long as 0a ?

(1) 5a (3) 5 2a

(2) 5a (4) 5

2

a

2) When a > 0 and b is a positive integer, the expression

( )2

1

5 ba is equivalent to

(~June ’16) 1)

( )2

1

5b a

(3) 1

5 ba

2) ( )5b

a (4) ( )

2

5b a

3) Which of the following equations represents exponential growth?

1) 1

2

x

y

=

(3) 100(1.04)xy =

2) ( )125 .75x

y = (4) 4xy = −

4) The population of fish in a local bay is modeled by the equation ( ) ( )2,500 1.025t

P t = ,

where t represents the number of years since 2,500 fish were first measured. Based on this equation, which of the following represents the percent the population will grow in a decade?

(1) 2.5% (3) 28% (2) 25% (4) 32%

Unit 4A Test Review

5) If $850 is invested in an account that earns 2.6% yearly interested compounded

quarterly, then which of the following is the worth of this investment after 5 years? (1) $960.50 (3) $967.60 (2) $966.40 (4) $968.25

6) The expression

2

3

1

x

is equivalent to

(1) 1x− (3) 5x−

(2) 2

3x (4) 6x−

7) Given the graph of the exponential function ( )x

y a b= shown below, which of the

following statements must be true? (1) 10 and 0a b

(2) 1 and 10a b

(3) 10 and 1a b

(4) 10 and 1a b

8) The value of x in the equation

2 5 34 8x x+ = is (June ’11)

1) 1 2) 2 3) 5 4) -10

9) Which of the following is the solution to the equation shown below in terms of the

unknown constant a?

5

2 18

2

a

x a− =

(1) 1

3x a= (3)

1

2x a= −

(2) 5

2x a= (4) x a= −

10) What is the function for the following table? (1) y = 2(4)x (3) y=16(1/4)x (2) y = 64(1/4)x (4) y= 16(2)x

11) Selected values of an exponential function of the form ( )x

y a b= are shown below.

To the nearest hundredth, which of the following represents the value of b? (1) 1.61 (3) 3.23 (2) 2.29 (4) 1.83

x 0 1 2 3 4

y 16 4 1 1/4 1/16

x 4 8 12 15 17

y 21 140 916 3754 9610

12) If a credit card company charges 13.5% yearly interest which expression can be used to approximate their monthly percent increase? [Let m represent months]

(1) ( )1.0106m

(3) (1.135)m

(2) ( ) 121.0106m

(4)

12

(1.135) m

13) How can you determine if an exponential equation represents exponential growth?

1) The initial value is negative 2) The base, b, is 0< b< 1 3) The base is a negative number 4) The base is a positive number greater than 1.

14) Which equation could be used to determine the value of an account that is

continuously compounded with an $18,000 initial investment, at an interest rate of 1.25% for 24 months? (June ’14)

1) 1.25 218,000A e=

2) 1.25 2418,000A e=

3) 0.0125 218,000A e=

4) 0.0125 2418,000A e=

15) Which of the following would give the same result as 3 a ?

(1) 1

3a (3) 6a

(2) 1

6a (4) 1

4a

16) If a = 4 and b= -3, what is the value of the expression

2

3

a

b

−

−

? (June ’10) 1) 16

27−

3) 27

16−

2) -1 4) 27

16 17) The half-life of a certain radioactive material is 82 hours. An initial amount of the material has a mass of 157 kg. Which of the following equations could be used to calculate the number of grams remaining from the original sample t hours after the scientist collected it?

18) In an exponential function, ( ) ( )x

f x a b= , it is known that ( )5 15f = and

( )8 170f = . Find b to the nearest hundredth.

(1) 1.87 (3) 3.19 (2) 2.25 (4) 3.72

1) 821 1

2 157

t

y

=

2) 821

1572

t

y

=

3) 821

2157

t

y

=

4) 821

1572

t

y

=

19) The population of fish in a local bay is modeled by the equation ( ) ( )2,500 1.025t

P t = ,

where t represents the number of years since 2,500 fish were first measured. Based on this equation, which of the following represents the percent the population will grow in a decade?

(1) 2.5% (3) 28% (2) 25% (4) 32% 20) If the population of Ashmore, Illinois is decreasing by 5.8% per year, then by what

percent will it decrease in the next 5 years? Show how you arrived at your result. Round to the nearest tenth of a percent.

21) A house purchased 5 years ago for $100,000 was just sold for $135,000. Assuming exponential growth, approximate the annual growth rate, to the nearest percent. (June ’16)

22) Francisco's calculator tells him that 1

29−

is equal to 1

3. Using properties of

exponents, explain this equivalence.

23) If ( )2 7

13

5

x

g x

+

= −

then algebraically determine the solution to the equation

( ) 22g x = .

24) Half-life is defined as the amount of time, often in years, that it takes for a radioactive

substance to loose half of its radioactivity. If a substance has a half-life of 58 years and starts with 500 grams radioactive, then how much remains radioactive after 30 years? Round to the nearest gram.

25) An exponential function, f(x), passes through the points (3,6) and (7,18). Write the equation of the exponential function to the nearest hundredth.

Determine the value of f(20) to the nearest integer. Only an algebraically based solution will receive full credit.

26) If $500 is placed in a savings account that earns a 6% nominal interest

compounded monthly, then which of the following represents the account’s worth after 10 years?

(1) $800.00 (3) $895.42 (2) $873.29 (4) $909.70

27) A population of fruit flies is increasing at a rate of 22.5% per hour. If the population had an original size of 10 flies, then which of the following is its size after one day?

(1) 798 (3) 1122 (2) 935 (4) 1304 28) The water level in a draining reservoir is changing such that the depth of water

decreases by 7.5% per hour. If the water starts at a depth of 45 feet, then which of the following functions properly models the depth, d, as a function of time, t, in hours since it started draining?

(1) ( )45 .075t

d =

(2) ( )45 .925t

d =

(3) ( )45 7.5t

d =

(4) ( )45 92.5t

d =